Chapter 2 Surfaces with Constant Mean Curvature


 Brianna McDaniel
 1 years ago
 Views:
Transcription
1 Chapter 2 Surfaces with Constant ean Curvature In this chapter we shall review some basic aspects of the theory of surfaces with constant mean curvature. Rather the reader should take this chapter as a first introduction to the problems and as a way to become acquainted with the methods that will be needed in successive chapters. In the process of this review, we shall obtain results on compact cmc surfaces with boundary. The surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and we shall relate it to the classical isoperimetric problem. Then we state the first and second variation formula for the area and we give the notion of stability of a cmc surface. Next, we shall introduce complex analysis as a basic tool in the theory and we define the Hopf differential. This will allow us to prove the Hopf theorem. In addition, we compute the Laplacians of some functions that contain geometric information of a cmc surface. As the Laplacian is an elliptic operator, we are in position to apply the maximum principle and we will obtain height estimates of a graph of constant mean curvature. Finally, and with the aid of the expression of these Laplacians, we will derive the Barbosado Carmo theorem that characterizes a round sphere within the family of closed stable cmc surfaces of Euclidean space. 2.1 The Variational Formula for the Area Let be a connected orientable (smooth) surface with possibly nonempty boundary. Denote by int() = \ the set of interior points of. Letx : R 3 be an immersion of in Euclidean threespace R 3. We say that is immersed in R 3 if the immersion x is assumed. If p, we write p instead of x(p),orsimply,x. We represent by N : S 2 a Gauss map (or an orientation), where S 2 denotes the unit sphere of R 3. An immersion x is called an embedding if x : x() is a homeomorphism and we say that the surface is embedded in R 3.If is compact, this is equivalent to x() having no selfintersections and we will identify the surface with its image x(). Let Γ R 3 be a (smooth) space curve. Consider an immersion x : R 3 of a surface. We say that Γ is the boundary of the immersion x if x : Γ is R. López, Constant ean Curvature Surfaces with Boundary, Springer onographs in athematics, DOI 1.17/ _2, SpringerVerlag Berlin Heidelberg
2 14 2 Surfaces with Constant ean Curvature a diffeomorphism. In particular, we prohibit that x turns many times around Γ. We also say that x (or ) spans Γ or that Γ is the boundary of x(). Ifx is an embedding we will use interchangeably and Γ. We endow with the Euclidean metric, induced by the immersion x. If is a compact surface, the areaof is defined by A = d, where d denotes the area element of. Now we introduce the volume of the immersion and we motivate the definition by considering a closed embedded surface. In such case, defines a bounded 3domain W R 3. We call the volume V enclosed by the Lebesgue volume of W. An expression of V in terms of an integral on may be derived by an easy application of the divergence theorem. The divergence of the vector field Y(x,y,z) = (x,y,z)is 3 and thus V = 1 = 1 Div(Y ) = 1 d = W 3 W 3 N,Y 1 N,x d, 3 where N is the Gauss map on that points towards W. In general, if x : R 3 is an immersion of a compact surface with possibly nonempty boundary, the algebraic volume of x with respect to the orientation N is defined by V = 1 N,x d. (2.1) 3 When, the number V measures the algebraic volume determined by the surface together with the cone C formed by and the origin of R 3. This follows if we parametrize C as φ(s,t)= tα(s), with α : x( ) a parametrization of x( ) and t 1. Since C together with forms a 2cycle, the same vector field Y gives 3V = N,x d N C,φ dc, C where N C is the unit normal vector to C. Finally, observe that N C,φ =. We remark that the volume of the immersion depends on the origin, or in other words, the number V changes by translations of the initial surface. However, when the boundary is planar, and as a consequence of the divergence theorem again, we have: Proposition Let x : R 3 be an immersion of a compact surface. Assume that x( ) = Γ is contained in a plane P. Then the volume is invariant provided the origin lies in P. A particular case occurs when is the graph of a function u C 2 (Ω) C (Ω), where Ω R 2 is a bounded domain and u = on Ω. Choose on the usual
3 2.1 The Variational Formula for the Area 15 parametrization of a graph, namely, Ψ(x,y) = (x,y,u(x,y)) and the orientation given by N ( x,y,u(x,y) ) 1 = ( u x, u y, 1). (2.2) 1 + u 2 x + u2 y As d = 1 + u 2 x + u2 y dxdy,wehave V = 1 3 N,Ψ d = 1 3 Ω (xu x + yu y u) dx dy. Define on Ω the vector field Y(q)= u(q)q. Then Div(Y ) = xu x + yu y + 2u and using that u = on Ω, we find that (xu x + yu y + 2u) dx dy = u n,q ds =, Ω where n is the inward conormal vector of Ω along Ω. We conclude that V = udxdy. (2.3) Ω As was noted in Chap. 1, the mean curvature of the surface can be motivated if we consider the socalled isoperimetric problem. Isoperimetric problem: Among all compact surfaces in Euclidean space R 3 enclosing a given volume, find the surface of least area. If we modify the question and we only ask for those surfaces that are solutions of the isoperimetric problem up to the first order, the problem becomes the following: Variational problem: Characterize a compact surface in Euclidean space R 3 whose area is critical among all variations that preserve the volume of the surface. Definition A variation of an immersion x : R 3 is a differentiable map X : ( ε, ε) R 3 such that for each t ( ε, ε), the maps x t : R 3 given by x t (p) = X(p,t) are immersions for all t, and for t =, x = x. The variational vector field of the variation {x t } is defined by ξ(p) = X(p,t) t Ω, p. t= The variation {x t } is said to be admissible if preserves the boundary of x, that is, x t (p) = x(p) for every p. In particular, ξ = on. Assume is a compact surface. Consider the area A(t) and the volume V(t)of induced by the immersion x t : A(t) = d t, V(t)= 1 N t,x t d t, 3
4 16 2 Surfaces with Constant ean Curvature where d t and N t stand for the area element of induced by x t and the unit normal field of x t, respectively. In particular, A() and V() are the area and the volume of the initial immersion x. We obtain the variation formula for A(t) and V(t)at t =. Proposition (First variation for the area) The area functional A(t) is differentiable at t = and A () = 2 H N,ξ d ν,ξ ds, (2.4) where ν is the inward unit conormal vector of along and H is the mean curvature of the immersion. The mean curvature H is defined by H(p)= κ 1(p) + κ 2 (p), p, 2 where κ 1 (p) and κ 2 (p) are the principal curvatures of x at p, i.e., the eigenvalues of the Weingarten endomorphism A p = (dn) p : T p T p. In particular, the sign of H changes by reversing the orientation N. On the other hand, the second fundamental form of the immersion is σ p (u, v) = (dn) p (u), v, u,v T p. Then the value 2H(p) agrees with the trace of σ p. Recall that the norm σ of σ is σ 2 = 4H 2 2K, where K = κ 1 κ 2 is the Gauss curvature of the surface. oreover, σ 2 2H 2 on and equality holds at a point p if and only if p is umbilical. Proof of Proposition A proof of (2.4) appears in Proposition A..1, Appendix A. Here we restrict to the case when the variational vector field is perpendicular to the surface. Let x t : R 3 be the variation of x given by x t (p) = x(p) + tf (p)n(p), t ( ε, ε) with f C (). The variational vector field is normal to the surface because ξ = fn. A computation leads to (dx t ) p (v) = v + tf (p) dn p (v) + tdf p (v)n(p), for v T p. By the compactness of, we can choose ε> sufficiently small in order to obtain (dx t ) p (v) for all tangent vectors v and thus, x t is an immersion. Let e i denote the principal directions at p. Then The Jacobian of x t is (dx t ) p (e i ) = ( 1 tκ i (p)f (p) ) e i + tdf p (e i )N(p).
5 2.1 The Variational Formula for the Area 17 Jac(x t )(p) = (dx t ) p (e 1 ) (dx t ) p (e 2 ) = t ( 1 tκ 2 (p)f (p) ) df p (e 1 )e 1 t ( 1 tκ 1 (p)f (p) ) df p (e 2 )e 2 ( 1 2tH (p)f (p) + t 2 K(p)f (p) 2) N(p). Hence it follows that d dt Jac(x t )(p) = 2H(p)f (p). (2.5) t= Finally, by (2.5) and the formula for change of variables, we find that A d () = dt Jac(x t )d = 2 Hf d = 2 H N,ξ d, t= showing (2.4). Observe that the boundary integral in (2.4) vanishes because ξ(p) T p. We say that is a surface with constant mean curvature if the function H is constant. We abbreviate this by saying that is a cmc surface or an H surface if we want to emphasize the value H of the mean curvature. In the particular case that H = on, we say that is a minimal surface. In view of Proposition 2.1.3, we obtain the following characterization of a minimal surface: Theorem Let x : R 3 be an immersion of a compact surface. Then the immersion is minimal if and only if A () = for any admissible variation of x. Proof If H =, then (2.4) gives immediately A () =. Conversely, let f C () be a function with f>inint() and f = on. Define the variation x t (p) = x(p) + tf (p)h (p)n(p), which is admissible and ξ = fhn. Then (2.4) yields = A () = 2 H N,ξ d = 2 fh 2 d. As f is positive on int(), then H = on. This theorem establishes that a compact surface is minimal if and only it is a critical point of the area for any admissible variation. As a consequence, if Γ is a closed curve and is a surface of least area among all surfaces spanning Γ, then is a minimal surface since for any variation of, the functional A(t) has a minimum at t =, hence, A () =. The reverse process does not hold in general and there exist compact minimal surfaces that are not minimizers. A special case is studied in Proposition below.
6 18 2 Surfaces with Constant ean Curvature Proposition (First variation for the volume) The volume functional V(t) is differentiable at t = and V () = N,ξ d. (2.6) For a proof, see Proposition A..2, Appendix A. Once the formulas for the first variation for the area and for the volume have been obtained, we have the conditions for characterizing a surface with constant mean curvature. If we recall the motivation of cmc surfaces by soap bubbles given in the preceding chapter, we require that the perturbations made on the soap bubble keep the enclosed volume of air constant. Thus we give the next definition. A variation is said to be volume preserving if the functional V(t) is constant. In particular, V (t) =. Comparing with the proof of Theorem for minimal surfaces and viewing the integral in (2.6), for a cmc surface it does not suffice that f = on. For this reason, we need the following result [BC84]. Lemma Let x : R 3 be an immersion of a compact surface and let f C () with fd=. Then there exists a volume preserving variation whose variational vector field is ξ = fn. Furthermore, if f = on, then the variation can be assumed admissible. Proof Let g be a differentiable function on such that g = on and gd. If I = ( ε, ε), define X : I I R 3, X(p,t,s)= x(p) + ( tf (p) + sg(p) ) N(p). For ε> sufficiently small, the function X may be seen as a variation of x fixing s or fixing t, with X(p,, ) = x(p).letv(t,s)be the volume of the surface X(,t,s): R 3 and consider the equation V(t,s)= c, where c is a constant. By (2.6), V(t,s) s = (t,s)=(,) gd. The implicit function theorem guarantees the existence of a diffeomorphism ϕ : I 1 I 2, where I 1 and I 2 are open intervals around, such that ϕ() = and V(t,ϕ(t))= c for all t I 1. This allows us to consider the volume preserving variation of x given by x t (p) = X(p,t,ϕ(t)). We show that ξ = fn. The derivative of V(t,ϕ(t))= c with respect to t yields = V t () + ϕ () V ( () = f + ϕ ()g ) d = ϕ () gd. s
7 2.1 The Variational Formula for the Area 19 This says that ϕ () =. Thus ξ(p) = x t t = ( f(p)+ ϕ ()g(p) ) N(p)= f (p)n(p). t= In the particular case that f = along, and since g = on,wehavex t (p) = x(p) along and thus the variation is admissible. Theorem Let x : R 3 be an immersion of a compact surface. Then x has constant mean curvature if and only if A () = for all volume preserving admissible variations. Proof Assume that the mean curvature H is constant. For a volume preserving variation of x, V (t) =. In addition, if the variation is admissible, ξ = on. Since H is constant, the expression of A () in (2.4)gives A () = 2H N,ξ d = 2HV () =. Conversely, assume that x is a critical point of the area for any volume preserving admissible variation. Let H = 1 Hd, A where A is the area of and define the function f = H H. Observe that H is constant if and only if f =. By contradiction, assume that at an interior point, f. As fd =, the sets + ={p int() : f(p)>} and ={p int() : f(p)<} are nonempty. Fix points p + +, p and consider the corresponding bump functions ϕ +,ϕ : R, that is, smooth nonnegative functions with supp(ϕ + ) +, supp(ϕ ) and ϕ + (p + ) = ϕ (p ) = 1. In particular, ϕ +,ϕ = on.let α + = ϕ + fd>, α = ϕ fd< and λ> be the real number such that α + + λα =. Define ϕ = ϕ + + λϕ.this nonzero function satisfies ϕ onint(), ϕ = on and ϕf d = ϕ + fd+ λ ϕ fd=. (2.7) + From the preceding lemma, there exists an admissible variation of x that preserves the volume and whose variational vector field is ξ = ϕf N. Then (2.4) and (2.7)now give = A () = 2 Hϕf d = 2 (H H )ϕf d = 2 ϕf 2 d.
8 2 2 Surfaces with Constant ean Curvature Thus ϕf 2 = on. However, at the point p +, (ϕf ) 2 (p + ) = f(p + ) 2 >, a contradiction. For cmc graphs, we have: Proposition Let Ω R 2 be a bounded domain and let be a compact cmc graph on Ω. Then minimizes the area among all graphs spanning and with the same volume as. Proof Suppose that is an H graph of a function u defined on Ω and let be another graph with = and the same volume as. Then determines an oriented 3domain W R 3 with zero volume V because the volumes of and agree. On W define the vector field Y(x,y,z) = N(x,y,u(x,y)), where N is the unit normal field of according to (2.2). We compute the divergence of Y. Taking into account (1.5) we see that ( ) ( ) ( ) u x u y 1 Div(Y ) = x + y + z 1 + u u u 2 u = div = 2H. 1 + u 2 The divergence theorem gives = 2HV = W = area() + Y,N d, Div(Y ) = Y,N d + Y,N d where N is the unit normal of pointing outwards from W. Hence, and as Y,N 1, we conclude area() = Y,N d d = area ( ). In fact, a more general result asserts that a minimal graph over a convex domain is areaminimizing among all compact surfaces with the same boundary [Fed69, or95]. Since a surface is locally the graph of a function defined in a convex domain, then we conclude that a minimal surface locally minimizes area. There exists another variational characterization of a surface with constant mean curvature using Lagrange multipliers. Given a (not necessarily volume preserving) admissible variation X and λ R, define the functional J λ as J λ (t) = A(t) 2λV (t), t ( ε, ε). (2.8)
9 2.1 The Variational Formula for the Area 21 By (2.4) and (2.6), J λ () = 2 (H λ) N,ξ d. (2.9) Theorem Let x : R 3 be an immersion of a compact surface. Then x has constant mean curvature H if and only if J H () = for all admissible variations. Proof If x has constant mean curvature H,takeλ = H in (2.9). To prove the converse, assume that there exists a λ R such that J λ () = for any admissible variation. In particular, it holds for any volume preserving admissible variation {x t }, and since λ is a constant, we have = J λ () = A () 2λV () = A (). Since A () = holds for all volume preserving admissible variation, we apply Theorem concluding that H is constant. This characterization of a cmc surface is related to the Plateau problem (1.2), where the solutions are obtained by minimizing D H in a suitable space of immersions from a disk into R 3, as was remarked in Chap. 1. Comparing the expressions of J λ and D H in (2.8) and (1.4) respectively, the area integral is replaced by the Dirichlet integral and the volume V by its formulation for a parametric surface. Among all critical points of the area, we consider those that are local minimizers. Then it is natural to study the second variation of the area because for a surface of least area, the second derivative A () is not negative. Since a surface with constant mean curvature is characterized in variational terms by Theorems and 2.1.9, there are two different notions of stability. Definition Let be a compact surface and let x : R 3 be an immersion of constant mean curvature H. 1. The immersion x is said to be stable if A () for all volume preserving admissible variations. 2. The immersion x is said strongly stable if J H () for all admissible variations. With respect to the first definition, some authors prefer to use the term volume preserving stable rather than stable. Proposition (Second variation for the area) Let be a compact surface and let x : R 3 be an immersion of constant mean curvature H. If {x t } is a volume preserving admissible variation with f = N,ξ, then A () = f ( f + σ 2 f ) d. (2.1) Here denotes the LaplacianBeltrami operator on.
10 22 2 Surfaces with Constant ean Curvature For a proof, see Corollary A..4 in Appendix A. The term in the parentheses of (2.1), namely, L(f ) = f + σ 2 f, is called the Jacobi operator of the immersion. As a consequence, the solutions of the isoperimetric problem are stable because they are minimizers of the area. Similarly, the formula for the second variation of the function J H is J H () = f ( f + σ 2 f ) d, for all admissible variation of the immersion. Here the function f only satisfies f = along. Let f be a smooth function on with f = on. Since div(f f)= f f+ f 2, the divergence theorem changes the expression of A () into A ( () = f 2 σ 2 f 2) d. (2.11) This allows us to extend the righthand side of (2.11) as a quadratic form I in the Sobolev space H 1,2 (), the completion of C () in L2 (): I : H 1,2 () R, I(f)= ( f 2 σ 2 f 2) d, where now f denotes the weak gradient of f. Therefore, a surface is: 1. stable if I(f) for all f H 1,2 () such that fd= ; 2. strongly stable if I(f) for all f H 1,2 (). A first example of a stable surface is the sphere. Proposition A round sphere is stable. Proof Assume without loss of generality that the sphere is S 2. Consider the spectrum of the Laplacian operator. The first eigenvalue is λ 1 = and the eigenfunctions are the constant functions. The second eigenvalue is λ 2 = 2[CH89]. By the Rayleigh characterization of λ 2, {S 2 = λ 2 = min 2 f 2 ds 2 S 2 f 2 ds 2 : f C ( S 2) }, fds 2 =. S 2 Hence S 2 f 2 ds 2 2 S 2 f 2 ds 2 for all differentiable function f with fd=. Since σ 2 = 2onS 2, this proves the stability of S 2. Proposition A planar disk and a spherical cap are stable. oreover, asmall spherical cap and a hemisphere are strongly stable but a large spherical cap is not strongly stable.
11 2.1 The Variational Formula for the Area 23 Proof If is a planar disk, then σ 2 = and I(f)= f 2 d, showing that is strongly stable. Consider now a spherical cap. Without loss of generality, assume that the radius of the spherical cap is 1. Denote by S + a hemisphere of S 2. We use the spectrum of the Laplacian with Dirichlet conditions, that is, the solutions of the eigenvalue problem f + λf = on, f = on. The first eigenvalue for the Dirichlet problem in S + is λ 1 (S + ) = 2. Now we employ the monotonicity property of λ 1 :if 1 2, then λ 1 ( 1 )>λ 1 ( 2 ) [CH89]. 1. Consider f C () with f = on and fd=. Extending f to S2 by, we obtain a function f H 1,2 (S 2 ) with S 2 fds 2 =. As S 2 is stable by the preceding proposition, I(f)= I( f). 2. Since a small spherical cap is included in a hemisphere S +, λ 1 () > λ 1 (S + ) = 2. Thus if is a small spherical cap or a hemisphere, the variational characterization of λ 1 () gives 2 λ 1 () f 2 d f 2 d for all f C (). As σ 2 = 2on, we conclude I(f). For a large spherical cap, λ 1 () < λ 1 (S + ) = 2. If f is an eigenfunction of λ 1 (), then f 2 d f 2 d = λ 1() < 2, so I(f)< and is not strongly stable. As we observe in the above proof, strongly stability is related to the eigenvalue problem associated with the Jacobi operator L = + σ 2, that is, { L(f ) + λf = on f = on. (2.12) The operator L is elliptic and thus its spectrum has many properties (see Lemma 8.1.3). For example, the first eigenvalue λ 1 (L) is characterized by { λ 1 (L) = min fl(f)d } f 2 d : f C (), f = on. Then we have immediately from (2.1): Corollary A compact cmc surface is strongly stable if and only if λ 1 (L).
12 24 2 Surfaces with Constant ean Curvature 2.2 The Hopf Differential In this section we associate to each cmc surface a holomorphic 2form, the socalled Hopf differential, that informs us about the distribution of the umbilical points on the surface. Let x : R 3 be an immersion of an orientable surface. We use the classical notation {E,F,G} and {e,f,g} for the coefficients of the first and second fundamental forms, respectively, in local coordinates x = x(u,v). Consider conformal coordinates (u, v) defined on an open subset V of, i.e., Take the Gauss map E = x u,x u =G = x v,x v, F = x u,x v =. N = x u x v x u x v, where is the cross product of R 3. The first and the second fundamental forms are, respectively,, = E ( du 2 + dv 2), σ = edu 2 + 2fdudv+ gdv 2, where e = N,x uu, f = N,x uv, g= N,x vv. The mean curvature H is given by H = e + g 2E. (2.13) In terms of the Christoffel symbols, the second derivatives of x are: On the other hand, x uu = E u 2E x u E v 2E x v + en x uv = E v 2E x u + E u 2E x v + fn (2.14) x uu = E u 2E x u + E v 2E x v + gn. N u = e E x u f E x v, N v = f E x u g E x v. (2.15) Using (2.13), (2.14) and (2.15), the Codazzi equations are e v f u = N v,x uu N u,x uv = E v 2E (e + g) = E vh, f v g u = N v,x uv N u,x vv = E u 2E (e + g) = E uh.
13 2.2 The Hopf Differential 25 By differentiating 2EH = e + g with respect to u and v,wehave 2E u H + 2EH u = e u + g u, 2E v H + 2EH v = e v + g v. With both expressions, the Codazzi equations now read as (e g) u + 2f v = 2EH u, (e g) v 2f u = 2EH v, (2.16) respectively. Let us introduce the complex notation z = u + iv, z = u iv and Define Equations (2.16) are then simplified as z = 1 2 ( u i v ), z = 1 2 ( u + i v ). Φ(z, z) = e g 2if. Φ z = EH z. (2.17) We point out that the zeroes of Φ are the umbilical points of the immersion since Φ(p) = if and only if e = g and f = atp. We express Φ as follows. Let us consider the derivatives x z and N z : Then and thus x z = 1 2 (x u ix v ), N z = 1 2 (N u in v ). x z,n z = 1 4 (e g 2if ) = 1 Φ(z, z), (2.18) 4 Φ(z, z) = 4 x z,n z. We study Φ under a change of conformal coordinates w = h(z), where h is a holomorphic function. Then x z = h (z)x w, N z = h (z)n w and Hence Φ(z, z) = 4 x z,n z = 4h (z) 2 x w,n w =h (z) 2 Φ(w, w). Φ(w, w)dw 2 = Φ(w, w)h (z) 2 dz 2 = Φ(z, z)dz 2. This equality means that Φdz 2 defines a global quadratic differential form on the surface. Definition The differential form Φdz 2 is called the Hopf differential.
14 26 2 Surfaces with Constant ean Curvature Theorem A conformal immersion x : R 3 has constant mean curvature if and only if Φ(dz) 2 is holomorphic. In such case, either the set of umbilical points is formed by isolated points, or the immersion is umbilical. Proof Given a conformal parametrization x(u,v), Eq.(2.17) givesφ z =, which is equivalent to saying that Φ is holomorphic on V. Thus the umbilical points agree with the zeroes of a holomorphic function. This means either Φ = onv or the umbilical points are isolated. In the first case, an argument of connectedness proves that the set of umbilical points is an open and closed set of and so is an umbilical surface. Theorem is also valid in hyperbolic space H 3 and for the sphere S 3. A direct consequence is the Hopf theorem [Hop83]: Theorem (Hopf) The only closed cmc surface of genus in the space forms R 3, H 3 and S 3 is the standard sphere. Proof Since the genus of is zero, the uniformization theorem says that its conformal structure is conformally equivalent to the usual structure of C. This is defined by the parametrizations z in C and w = 1/z in C {}. The intersection domain of both charts is C {} and in this open set, the Hopf differential Φ is Φ(z) = w (z) 2 Φ(w) = 1 z 4 Φ(w). It follows that ( ) 1 lim Φ(z) = lim z z z 4 Φ(w = ) =. This shows that by writing Φ = Φ(z) in terms of the parametrization z on C, Φ can be extended to by letting Φ( ) =. Therefore Φ is a bounded holomorphic function on C. Liouville s theorem asserts that the only bounded holomorphic functions on C are constant. As Φ( ) =, then Φ = onc and all points are umbilical. Finally, the only umbilical closed surface in a space form is the sphere. 2.3 Elliptic Equations for cmc Surfaces Let x : R 3 be an immersion of a surface and let N be the Gauss map. Denote by X() the space of tangent vector fields on. Recall the Gauss and Weingarten formulae: X Y = XY + σ(x,y)n, (2.19) X N = A NX = AX, σ (X, Y ) = AX, Y, (2.2)
15 2.3 Elliptic Equations for cmc Surfaces 27 for X, Y X().Here is the usual derivation of R 3, is the induced connection on and A is the Weingarten map. We calculate the Laplacian of the position vector x and the Gauss map N. In order to compute these Laplacians, we use the fact that, fixing p, there exists a geodesic frame around p, that is, an orthonormal frame {e 1,e 2 } around p such that at p, ( ei e j ) p = (for a proof see, for instance, [Car92, Chv94]). Then ( ei e j ) p is orthogonal to at p and if f is a smooth function on, the Laplacian of f is f (p) = 2 e i e i, f = 2 ( e i ei (f ) ). (2.21) The next result is valid for any immersion without the need that H is constant. Proposition Let x : R 3 be an immersion of a surface. If a R 3, then x,a =2H N,a (2.22) x 2 = 4 + 4H N,x. (2.23) Proof Fix p and let {e 1,e 2 } be a geodesic frame around p. Using(2.19) and (2.21), we obtain x,a = = 2 ( e i ei x,a ) = 2 e i e i,a = 2 σ(e i,e i ) N,a =2H N,a. 2 ei e i,a Equation (2.23) derives from (2.22) and the properties of the Laplacian. Indeed, let {a 1,a 2,a 3 } be an orthonormal basis of R 3.Letx i = x,a i and N i = N,a i.as x i 2 = 1 N 2 i,wehave x 2 = 3 x i x i x i 2 = 4H 3 x i N i ( ) 1 N 2 i = 4H N,x +4. Now assume that the mean curvature H is constant. Proposition Let x : R 3 be an immersion with constant mean curvature. Then N,a + σ 2 N,a =. (2.24) N,x = 2H σ 2 N,x. (2.25)
16 28 2 Surfaces with Constant ean Curvature Proof Fix p and let {e 1,e 2 } be as above. Using (2.19) and (2.2) and because 2H = trace(a), we know that 2H = 2 2 Ae i,e i = e i N,e i is constant on. Thus for j {1, 2} we find that = e j ( 2 = e i N,e i ) = 2 ej e 2 i N,e i + ei N, e j e i 2 ei e 2 j N,e i = e i ej N,e i. (2.26) Now we compute the Laplacian of N,a using (2.21): N,a = = = 2 e i ei N,a = 2 ( e i ei N,e j a,ej ) i,j=1 2 ( e i ej N,e i a,ej ) i,j=1 2 i,j=1 e i ( ej N,e i ) a,ej + 2 ( = ej N,e i ei a,e j ) = = i,j=1 2 ( ej N,e i ei a,e j ) i,j=1 2 ( ) ej N,e i a, ei e j i,j=1 2 σ(e i,e j ) 2 N,a = σ 2 N,a. i,j=1 We have used in (*) that v N = Av is a selfadjoint endomorphism; in the step (**) we apply (2.26). As in the proof of (2.23) and for the computation of the Laplacian of N,x, we express N,x = 3 N i x i with respect to an orthonormal basis of R 3 and use the properties of the Laplacian. Equations (2.22) and (2.24) generalize to the other space forms. Denote by 3 (c) the space forms H 3, R 3 or S 3 if c = 1, or 1, respectively. We view the threedimensional sphere S 3 as a submanifold of R 4 and the hyperbolic space H 3 as a submanifold of the 4dimensional Lorentzinkowski space L 4 (see Chap. 1). We have [KKS92, Rsb93]:
17 2.3 Elliptic Equations for cmc Surfaces 29 Proposition Let x : 3 (c) be an immersion of a surface and a R 4. Then x,a = 2c x,a +2H N,a. (2.27) If, in addition, the mean curvature H is constant, then N,a =2cH x,a σ 2 N,a. (2.28) Equations (2.22), (2.23), (2.24), (2.25) involve the Laplacian operator and thus are elliptic. oreover, the last two equations have been obtained using the property that the mean curvature is constant. This justifies the title of this section. Once we compute these Laplacians, we shall obtain some geometric results for a given cmc surface. The key ingredient is that for a linear elliptic equation, such as the equation for the Laplacian operator, there is a maximum principle. In our context of compact surfaces, if f is a smooth function on such that f (resp. ), then max f = max f (resp. min f = min f ) and if f attains a maximum (resp. minimum) at some interior point of, then f is constant [GT1]. The first result provides an estimate of the height of a compact cmc graph, usually called the Serrin estimate in the literature. This will be achieved by forming an appropriate linear combination of the functions x,a and N,a. We first need the following auxiliary result: Lemma Let be an H surface of R 3 and a R 3. Consider the function f = H x,a + N,a. Then f is constant if and only if is umbilical. Proof It is straightforward that if is an opensubsetof aplaneor asphere, then f is constant. Assume now that f is constant. By contradiction, suppose p is not umbilical. Then there exists an orthonormal frame {e 1,e 2 } of principal directions in an open set V around p, that is, e i N = κ i e i on V. Then = e i (f ) = (H κ i ) e i,a, i = 1, 2. Since there do not exist umbilical points, H κ i, i = 1, 2. We deduce that e i,a = in an open subset of V, i = 1, 2. This means that N = a in V and is part of a plane. In particular, p is an umbilical point, a contradiction. The proof of the next theorem has its origin in the use of Bonnet s parallel surfaces by H. Liebmann and later by J. Serrin [Ser69a]. Here we follow W.H. eeks III in [ee88]. Further generalizations can be found in [Lop7]. Theorem Let Ω R 2 be a bounded domain and H R. Let be an H  graph of R 3 of a function u C 2 (Ω) C (Ω). Let e 3 = (,, 1) and we orient so that N,e If H>, then min Ω u 1 H u max Ω u.
18 3 2 Surfaces with Constant ean Curvature In addition, if the equality holds at a point on the lefthand side, then is part of a sphere of radius 1/H. 2. If H<, then min Ω u u max Ω u 1 H. In addition, if the equality holds at a point on the righthand side, then is part of a sphere of radius 1/H. 3. If H =, then min Ω u u max Ω u. In the particular case H and u = along Ω, we have u 1/ H and equality attains at a point if and only if Ω is a round disk and is a hemisphere Proof Instead of working with the function u, we consider the surface as an embedding x : R 3. Recall that u is the height function u = x,e 3.As is a cmc surface and the boundary is an embedded analytic curve, then extends analytically across [ul2]. Suppose H. We give the proof for the case H> (the proof is similar if H<). By (2.22), x,e 3 > and by the maximum principle, x,e 3 max x,e 3. This proves the right inequality of item (1). Combining (2.22) and (2.24), jointly σ 2 2H 2, we obtain ( H x,e 3 + N,e 3 ) = ( 2H 2 σ 2) N,e 3. We apply the maximum principle: since N,e 3, we have ( H x,e 3 + N,e 3 min H x,e3 + N,e 3 ) H min x,e 3. As H>, x,e 3 N,e 3 H + min x,e 3 1 H + min x,e 3, (2.29) proving the left inequality of item (1). If at a point p, x(p),e 3 = 1/H + min x,e 3, then we conclude by (2.29) that N(p) = a and p int(). Then the function f = H x,e 3 + N,e 3 attains a minimum at an interior point and this implies that f is a constant function. Lemma proves that is umbilical and as H, is part of a sphere. Assume H = and u = on. Then (2.22) yields x,e 3 = and the estimates follow from the maximum principle. If H, u = on and at a point, u =1/ H, then is part of a sphere with planar boundary. But the estimate is sharp if is a hemisphere. Observe that for small values of H, the above estimates are not good in the following sense. Assume that u = on Ω. Then the graph of u = onω is a minimal surface and for values near to H =, there exist H graphs on Ω spanning
19 2.3 Elliptic Equations for cmc Surfaces 31 Ω. By continuity, these H graphs are close to the plane z = ofr 3 and thus their heights are small. However, the number 1/ H is very large. When the domain is not bounded, a generalization of the estimates for H graphs will be given in Theorem and Corollary Remark Similar arguments to those used in Theorem will be used later in different contexts to obtain height estimates provided we have elliptic equations such as (2.22) and (2.24) in order to use the maximum principle. Here we recall two settings. First, consider a hypersurface n in a space form n+1 (c). Denote by κ 1,...,κ n the principal curvatures of and let S r = i 1 < <i r κ i1 κ ir.the rmean curvature H r of is defined by ( ) m H r r = S r. We remark that H 1 coincides with the mean curvature H of, n(n 1)H 2 is the scalar curvature and H n is the GaussKronecker curvature of. H. Rosenberg obtained height estimates for a compact graph n on a geodesic hyperplane of R n+1 or H n+1 provided for some r, H r+1 is a positive constant [Rsb93]. The second setting is the class of compact surfaces in Euclidean space with positive constant Gauss curvature K. If is a such surface, the second fundamental form σ defines a Riemannian metric on. With respect to this metric, we compute the Laplacian of the height function and the Gauss map, obtaining elliptic equations by the positivity of K [Ga]. Then one can obtain height estimates for a graph with constant Gauss curvature K> on a domain of a plane or a radial graph on a domain of a sphere [Gb, Lop3b]. The second result that we derive is the Jellet theorem [Jel53], which refers to starshaped surfaces. A starshaped surface in R 3 is a compact embedded surface such that 3domain W R 3 that it bounds is starshaped in the affine sense, that is, there exists a point p W such that the segment joining p with any point of W is included in W. Theorem (Jellet) The only starshaped surface in R 3 with constant mean curvature is the round sphere. Proof Let be a starshaped cmc surface and suppose after a rigid motion that p is the origin of coordinates O = (,, ). The property of being starshaped is equivalent to the property that we cannot draw a straight line from O which is tangent to. This means that the support function based on the point O has constant sign on. Consider the orientation N that points towards W and denote by x the position vector of. Then the support function is N,x and because N is the inward orientation, we see that N,x < on. We apply the divergence theorem in (2.23). As H is constant, we find that A + H N,x d =, (2.3)
20 32 2 Surfaces with Constant ean Curvature where A is the area of. On the other hand, the divergence theorem in (2.25) yields = 2AH + ( = 2H A + H σ 2 N,x d 2AH + 2H 2 ) N,x d =, N,x d wherewehaveusedin( ) that σ 2 2H 2 and (2.3)in( ). Equality implies that σ 2 = 2H 2 on. Then is umbilical and this proves the theorem. Jellet s theorem, formulated in 1853, may be viewed as a precursor of the Hopf theorem established in 1951 [Hop83] because the genus of a starshaped surface is zero. Following this theme, another generalization of Theorem is Alexandrov s theorem, proven a century later, which asserts that a closed embedded cmc surface is a round sphere [Ale56, Ale62]. In this case, a starshaped surface is embedded. A third application of the computations of the above Laplacians is the following result. Theorem Planar disks and small spherical caps are the only cmc graphs spanning a circle. As we shall see in the next chapter, this result is trivial by the tangency principle (see Theorem 3.2.6). However, the purpose here is to give a proof that does not involve the tangency principle but uses Eqs. (2.22) and (2.24) [Lop9]. In a similar context and for a closed cmc surface, R. Reilly obtained another proof of Alexandrov s theorem without the use of the maximum principle thanks to a combination of the inkowski formulae [Rei82]. Proof Let Γ R 2 {} be a circle of radius r> about the origin and denote by Ω the domain bounded by Γ. Consider a compact H surface which is a graph of u defined on Ω and with u = on Ω. Then u satisfies (1.5) where H is computed by the unit normal vector given in (2.2). Then N,e 3 > on, where e 3 = (,, 1). Let α be the parametrization of Γ such that α ν = N, where ν is the inner unit conormal vector of along Γ. We know that α = α/r 2. Since N,e 3 > on, we see that α α = re 3 and ν,e 3 = N α,e 3 = N,α 1 e 3 = N,α. (2.31) r Using (1.5), we obtain 2πr 2 u H =, n = Ω 1 + u 2 ν,e 3 ds, (2.32)
14.11. Geodesic Lines, Local GaussBonnet Theorem
14.11. Geodesic Lines, Local GaussBonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationORIENTATIONS. Contents
ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit nsimplex which
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More information1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy and xzplanes, etc. are For example, z = f(x, y), x =
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationFINAL EXAM SOLUTIONS Math 21a, Spring 03
INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationNOTES ON MINIMAL SURFACES
NOTES ON MINIMAL SURFACES DANNY CALEGARI Abstract. These are notes on minimal surfaces, with an emphasis on the classical theory and its connection to complex analysis, and the topological applications
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationINVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS
INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the leftinvariant metrics with nonnegative
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More information4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complexvalued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complexvalued function of a real variable Consider a complex valued function f(t) of a real variable
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More information1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1
Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationExtrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal RodríguezBernal Alejandro VidalLópez Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationInvariant Metrics with Nonnegative Curvature on Compact Lie Groups
Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.
More informationTOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
More informationThe Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006
The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationtr g φ hdvol M. 2 The EulerLagrange equation for the energy functional is called the harmonic map equation:
Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationMathematical Physics, Lecture 9
Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential
More informationTHE HELICOIDAL SURFACES AS BONNET SURFACES
Tδhoku Math. J. 40(1988) 485490. THE HELICOIDAL SURFACES AS BONNET SURFACES loannis M. ROUSSOS (Received May 11 1987) 1. Introduction. In this paper we deal with the following question: which surfaces
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationA HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R
A HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R UWE ABRESCH AND HAROLD ROSENBERG Dedicated to Hermann Karcher on the Occasion of his 65 th Birthday Abstract. A basic tool in
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vectorvalued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More information