SURFACES OF CONSTANT MEAN CURVATURE


 Jeremy Poole
 1 years ago
 Views:
Transcription
1 SURFACES OF CONSTANT MEAN CURVATURE CARL JOHAN LEJDFORS Master s thesis 2003:E11 Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM
2
3 i Abstract The aim of this Master s dissertation is to give a survey of some basic results regarding surfacesëof constant mean ) 0 curvature (CMC) inê3. Such surfaces are often called soap bubbles since a soap film in equilibrium between two regions is characterized by having constant mean curvature. The surface area of these surfaces is critical under volumepreserving deformations. CMC surfaces may also be characterized by the fact that their Gauss map N :Ë S2 is harmonic i.e. it satisfies Ø(N whereø(n ) is the tension field of N, generalizing the classical Laplacian. This is a nonlinear system of partial differential equations. It was proved in the 1990s that this system has global solutions on compact surfaces of any genus g 0. In this dissertation we study necessary and sufficient conditions for a surface to have CMC. We study the minimal case (characterized by mean curvature H 0) and the wellknown Weierstrass representation for such surfaces. Also CMC surfaces with rotational symmetry are considered and a generalization of the Weierstrass representation to surfaces of nonzero constant mean curvature is presented. Finally we show that the only compact embedded CMC surfaces inê3 are spheres. It has been my intention throughout this work to give references to the stated results and credit to the work of others. The only part of this Master s dissertation which I claim is my own is the elementary proof of a special case of RuhVilms theorem for surfaces inê3 given in Theorem 4.1.
4 ii
5 iii Acknowledgements I wish to thank my supervisor, Sigmundur Gudmundsson, for his time, knowledge and patience. In particular, I wish to express my gratitude for him inspiring me to study the wonderful subject of geometry. Carl Johan Lejdfors
6
7 Contents Short History 1 Chapter 1. Some basic surface theory Notation Isothermal coordinates The tension field 7 Chapter 2. Minimal surfaces Conformality of the Gauss map The Weierstrass representation formula 10 Chapter 3. CMC surfaces of revolution Kenmotsu s solution Delaunay s construction 17 Chapter 4. CMC surfaces Harmonicity of the Gauss map Kenmotsu s representation formula 24 Chapter 5. Compact CMC surfaces 33 Recent developments 35 Appendix A. Harmonic maps 37 Bibliography 41 v
8
9 Short History In 1841 Delaunay characterized in [1] a class of surfaces in Euclidean space which he described explicitly as surfaces of revolution of roulettes of the conics. These surfaces are the catenoids, unduloids, nodoids and right circular cylinders. Today they are known as the surfaces of Delaunay and are the first nontrivial examples of surfaces having constant mean curvature, the sphere being the trivial case. In an appended note to Delaunay s paper M. Sturm characterized these surfaces variationally as the extremals of surfaces of rotation having fixed volume while maximizing lateral area. Using this characterization the following theorem was obtained: THEOREM (Delaunay s theorem). The complete immersed surfaces of revolution inê3 having constant mean curvature are exactly those obtained by rotating about their axis the roulettes of the conics. These surfaces where also recognized by Plateau using soap film experiments. In 1853 J. H. Jellet showed in [2] that ifëis a compact starshaped surface inê3 having constant mean curvature then it is the standard sphere. A hundred years later, in 1956, H. Hopf conjectured that this, in fact, holds for all compact immersions: CONJECTURE (Hopf s conjecture). LetËbe an immersion of an oriented, compact hypersurface with constant mean curvature H 0inÊn. ThenËmust be the standard embedded (n 1)sphere. Hopf proved the conjecture in [3] for the case of immersions of S 2 intoê3 having constant mean curvature and a few years later A. D. Alexandrov showed the conjecture to hold for any embedded hypersurface inên, see [4]. It was widely believed that this conjecture was true until 1982 when WuYi Hsiang constructed a counterexample inê4. Two years later Wente constructed in [5] an immersion of the torus T 2 inê3 having constant mean curvature. Wente s construction has been thoroughly studied but has only been able to create surfaces having genus g 1. A different method for constructing surfaces inê3 having constant mean curvature of any genus g 3 was presented in 1987 by N. Kapouleas [6]. A proof of the fact that there exist CMCimmersions of compact surfaces of any genus was published in [8] in 1995 by the same author. 1
10
11 CHAPTER 1 Some basic surface theory In this chapter we introduce the notation to be used in this text. We also introduce some basic results concerning isothermal coordinates and the tension field of the Gauss map of a surface inê Notation DEFINITION 1.1. A nonempty subsetëonê3 is said to be a regular surface if for each point p¾ëthere exists an open neighborhood U inë around p and a bijective map (x y): U Ë Ê2 that its inverse X : (U ) U such i. is a homeomorphism, ii. is a differentiable map, iii. (X x X y )(q) 0for all q¾ (U ). The functions x y are called local coordinates around p. The map X is called a local parametrization ofëaround p. LetËbe a regular surface inê3 and p¾ëbe an arbitrary point. By a tangent vector toë, at the point p, we mean the tangent vector ¼(0) of a differentiable parametrized curve : ( ) Ëwith (0) p. y p¾ë The set of tangent vectors ofëat a point p¾ëis called the tangent space ofëat and is denoted by T pë. A local parametrization X determines a basis X x X of T pë, called the basis associated with X. On the tangent plane we have the usual induced metric from the ambient spaceê3 with the associated quadratic form I p : T pë Êcalled the first fundamental form ofëat p¾ë. Given a local parametrization X ofëand a parametrized curve (t) X (x(t) y(t)) for t¾( ) p (0) we have y¼«2 with the following form I p ( ¼(0)) ÅX x x¼ X y X x X x p (x¼) 2 2ÅX x X y«p x¼y¼ ÅX y X y«p (y¼) 2 (1.1) E(x¼) 2 2Fx¼y¼ G(y¼) where the values of E, F and G are computed for t 0. By condition (iii) of the definition of a regular surface (1.1) we have, given a local parametrization X of a surfaceëinê3 at a point p¾ëthat the map y y¼ X x x¼ X 3
12 4 1. SOME BASIC SURFACE THEORY N :Ë S 2 defined by x X y N (p) X (1.2) X x X y (p) II p (v) ÅdN p (v) v«y¼ with p (0) X (x(0) y(0)), we get is well defined. This map is known as the Gauss map ofë. The quadratic form II p defined in T pëby is called the second fundamental form ofëat p. Given a local parametrization X onëat a point p¾ëand, as above, letting be a parametrized curve such that (t) X (x(t) y(t)) for t¾( ) dn p ( ¼) N¼(x(t) y(t)) N x x¼ N ( ¼) ( ¼) ¼«y Since N N 1 we must have that N x N y¾t y pëand hence N x a 11 X x a 21 x¼«x (1.3) N y a 12 X x a 22 X for some functions a ij. We find that II f p ÅdN p ÅN x x¼ N y y¼ X x x¼ X g e 2 v e(x¼) 2 2fx¼y¼ g(y¼) xx (1.4) where N x X x N X yy«yx«y«ån y X ÅN x«ån X xy«ån X x X ÅN y X y«ån X Using the terms from equations (1.1), (1.3) and (1.4) we arrive at eg a 11 ff 2 fg a 12 gf 2 EG F EG F fe ge a 21 ef a 22 ff EG F EG F known as the Weingarten equations. Continuing by using that X x X y is a basis for T pëand that N is orthogonal to both X x and X y we have that X x X y N is a basis forê3. We find that X xx 1 11 X x 2 11 X X y en yx 1 12 X x 2 12 X X xy 1 21 X x 2 21 X X y fn yy 1 22 X x 2 22 X y gn x 0 (1.5) The k ij are known as the Christoffel symbols and are invariant under isometries (i.e. can be computed from the first fundamental form alone). Using that (X xx ) y (X xy )
13 we find that y ISOTHERMAL COORDINATES 5 (X yy ) x (X xy ) N xy N yx ) g 2 11 e y f x e 1 12 f ( 2 12 f y g x e 1 22 f ( 2 22 E X x X x Ð2 ÅX K eg f 2 EG F 2 H eg 2fF ge 2(EG F 2 ) 1 12 ) g 2 12 (1.6) These equalities are known as the MainardiCodazzi equations. DEFINITION 1.2. LetËbe a surface inê3 and p¾ëan arbitrary point. Let dn p : T pë T pëbe the differential of the Gauss map. Then the determinant of dn p is called the Gaussian curvature K ofëat p. The negative half of the trace of dn p is called the mean curvature H ofëat p. In terms of the first and second fundamental forms K and H are given by (1.7) (1.8) 1.2. Isothermal coordinates In this section we introduce the notion of isothermal coordinates which is a useful tool in differential geometry. DEFINITION 1.3. LetËbe a surface y«g F 0 inê3. Then local coordinates (x y): U Ë Ê2 onëare said to be isothermal if there exists a strictly positive function, called the dilation,ð: U Ë Êsuch that y X We have the following result regarding existence of isothermal coordinates on an arbitrary surface inê2. THEOREM 1.4. LetËbe a differentiable surface inê3 and p¾ëbe a point onë. Then there exists an open neighborhood U of p and isothermal coordinates (x y): U Ë Ê2 2Ð2 around p. This was proved in the analytic case by Gauss. For a complete proof in the general case please see [9]. Having chosen isothermal coordinates the mean curvature simplifies 2fF H eg ge 2(EG F ) e g 2 The Christoffel symbols similarly 12 x simplify 22 1 y 2Ð2Ð (1.9) Ð2Ð
14 6 1. SOME BASIC SURFACE THEORY Using this we get the following form of the MainardiCodazzi equations (1.6): e y f x (e g) 2 g x f y (e g) (1.10) The Weingarten relations (eq. 1.3) reduce to a 11 e Ð2 a 21 a 12 fð2 a 22 gð2 (1.11) SupposeËis a surface inê3 and (x y): U Ë Ê2 are local isothermal coordinates onë. We may then consider the local coordinates (x y) as a complexvalued map z x iy : U Ë. The inverse X : z(u ) Ëcan then be considered as y map from an open subset z(u ) in intoëi.e. a local parametrization ofë. We then have X z 1 X x ix 2 U Ë The complex notation for surfaces inê3 has many advantages which we will be useful in chapters 2 and 3. Letting be the usual inner product inê3 and let ( ) be the complex bilinear extension of in 3 we have the following result. PROPOSITION 1.5. LetËbe a surface inê3 and let z x iy : be local isothermal coordinates onë. Then the inverse X : z(u ) Ëof z is conformal i.e. satisfies y«0 y«0 4 (X z X z ) X x 2 X y 2 2iÅX x X (1.12) 4 (X z X z) X x 2 X y 2 2iÅX x X Conversely, if z x iy are local coordinates onësatisfying equation U Ë (1.12) then they are isothermal. ) PROOF. The first statement follows by a direct computation. The reverse implication follows by considering real and imaginary parts of equation (1.12). PROPOSITION 1.6. LetËbe a surface inê3 and z x iy : be local isothermal coordinates onëwith dilationð. Then the inverse X : (U Ëof z satisfies where N :Ë S 2 4X zz X xx X yy 2Ð2 HN is the Gauss map ofë. PROOF. By a direct computation using the differentiated form of equation (1.12) we have 4X zz 2Ð2 [(X zz X z ) X z (X zz X z) X z] 4 (X zz N ) N ÅX xx X yy N«N 4(e g)n 2Ð2 HN
15 1.3. THE TENSION FIELD 7 This immediately gives our sought result The tension field In this section we give an explicit formula for the tension field (see appendix A) for maps from a surface inê3 into S 2 in terms of local isothermal coordinates. PROPOSITION 1.7. LetËbe a surface inê3 and :Ë S2 be a map into the unit sphere S 2 inê3. If (x y): U Ë Ê2 are local isothermal coordinates onëthen the tension fieldø( ) of is locally given by T Ø( ) 1Ð2 i.e. as the tangential part of the classical Laplacian x y 2 ek) inê2. PROOF. By the definition of the tension field of a smooth map :Ë S2 we have Ø( ) 2 k 1 d (ek) Ö ek d (Öek whereö is the pullback connection on the pullback bundle 1Ð2 1 TS overë 2 via. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be isothermal coordinates around p. We then have k Ð2 Öe k e k e k e k ( 2 2e 1Ð2 )e k g(e k e k ) grad Ö1Ðx 1Ðx Ö1Ðy 1Ðy 1Ð2 k Ð2 e k e k 1Ð2 ( 2 2e 1Ð2 )e k grad for k 1 2. Then using the definition of the gradient grad we obtain Öe 1 e 1 Öe 2 e 2 e 1 e 1 e 2 e 2 Ð2 e 1 ( 1Ð2 )e 1 e 2 ( 1Ð2 )e 2 grad ) 1Ðx 1Ð x 1Ðy 1Ð 1Ðx 1Ð x 1Ðy 1Ð y This implies that d (Öe 1 e 1 Öe 2 e 2 (1.13) y
16 ) 1Ðx 1Ð ) 1Ðy 1Ð y T 1Ð y 1Ð x T 1Ð x 1Ð y 1Ð SOME BASIC SURFACE THEORY The other term is given by Ö e 1 d (e 1 Ö e 2 d (e 2 1Ð T It follows by equations (1.13), (1.14) and (1.15) that 2 T 1Ð2 T x y Ø( ) Ö e 1 d (e 1 ) Ö e x 1Ð2 2 T 2 d (e 2 ) d (Öe 1 e 1 ) d (Öe 2 e 2 ) (1.14) (1.15) Harmonic maps generalize the concept of harmonic functions well known from complex analysis. A harmonic map is one for which the tension field vanishes everywhere and, as stated in Appendix A, arises as a critical point of a certain variational problem. THEOREM 1.8. LetËbe a surface inê3 and :Ë S2 be a map into the unit sphere Å xx y«s 2 inê3. If is conformal then it is harmonic. PROOF. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be xx x Å yx y«å yx x«å yy x«å yy y«å xy x«å xy y«local isothermal coordinates around p. Then the conformality of means that By differentiating we then obtain and thereforeå xx yy x«å yx yy«å xx yy y«thatø( ) 1Ð2 T 0 Å yx xx«å xy xx«0 Å yx yy«0 These relations imply Å x y«0 and x x Å y y«
17 CHAPTER 2 Minimal surfaces In this chapter we introduce some results concerning minimal surfaces. We also prove the famous Weierstrass representation for minimal surfaces. DEFINITION 2.1. A surfaceëinê3 is said to be minimal if its mean curvature H satisfies H Conformality of the Gauss map PROPOSITION 2.2. LetËbe a minimal surface inê3. Then the Gauss map N :Ë S ofëis conformal. PROOF. Let p¾ëbe an arbitrary point onëand (x y) be local isothermal coordinates around p. Then it follows by H e g y 2Ð2 0 y and equation (1.11) that N x 1Ð2 ex x fx N y 1Ð2 fx x 0 ex This implies that ÅN y«e2 f 2 x N y«0 and N y N Ð2 Hence N is conformal. A partial reverse implication of the previous theorem is obtained via the following. PROPOSITION 2.3. LetËbe a real analytic surface inê3 and N :Ë S2 be a Gauss map ofë. If N is conformal thenëis either minimal y or part of a sphere. PROOF. For local isothermal coordinates (x y) onëwe have ex x fx y 1Ð2 fx x gx 1Ð4 ef X e g 2fH y«x X x fgåx y X fð2 0 ÅN x N y«1ð2 x N x ÅN Let p¾ëbe a point. Suppose H(p) 0 then there exists an open neighborhood V Ëaround p such that f V 0. For every point in q¾v we have, 9
18 10 2. MINIMAL SURFACES by equation (1.11), that N x and N y are parallel with X x and X y, respectively. By x conformality we y have that N x N y and, since (x y) are isothermal, that X x X y. Hence q is umbilical i.e. the principal curvatures coincide. Let k k 1 k 2, where k 1 and k 2 are the principal curvatures. Differentiating N kx x and N kx y gives (kx x ) y (N x ) y (N y ) x (kx y ) x and since X x and X y N are linearly independent we must have k x k y 0 so k is constant. If k 0 then H 0 which contradicts 2 the assumption. Hence k 0 and kx a where a is a constant vector. Then X is a local parametrization for a sphere having radius 1 k centered at a k since 1 X a 2 1 N 2 1 k k k Thus by real analyticityëis either minimal or part of a sphere The Weierstrass representation formula The Weierstrass representation formula was first presented by Karl Weierstrass in [10]. It states that given two holomorphic functions defined on some simply connected subset of there exists an associated minimal surface. This surface is unique up to motions. THEOREM 2.4. LetËbe a surface inê3 and x iy : U Ë be local isothermal coordinates onë. Suppose U is an open simply connected subset ofë. If X : z(u ) Ëis the inverse of z thenëis minimal if and only if the derivative X z U Ë : z(u ) 3 is holomorphic. PROOF. This is a direct consequence of Proposition 1.6 and the fact that a map f : U is holomorphic if and only if f z 0. Integration gives us the following corollary. COROLLARY 2.5. LetËbe a surface inê3 and x iy : be local isothermal coordinates onë. Suppose U is an open simply connected subset ofë. Then the inverse X : z(u ) Ëof z is given by X (z) 2 Re z X z z 0 (z)dz C (2.1) where C is some constant vector inê3. PROOF. We have X z dz 1 (X x ix y )(dx idy) dx dx 2 1 X x dx X y dy i(x x dy X y 2 X zd z 1 X x dx X y dy i(x x dy X y 2
19 2.2. THE WEIERSTRASS REPRESENTATION FORMULA 11 Integrating dx X z dz X zd z 2 Re X z dz gives us our sought relation. This corollary gives us the famous Weierstrass representation for minimal surfaces. THEOREM 2.6 (Weierstrass Representation). Let V be an open simply connected subset of. Suppose f : V is holomorphic on V, g : V is meromorphic on V and the product fg 2 is holomorphic on V. Then X : V Ê3 defined by X (z) Re z X z z 0 (z)dz (2.2) where X z (z) f(z)(1 g(z) 2 i(1 g(z) 2 ) 2g(z)) is a minimal surface. PROOF. Using the above results the only thing we need to show is that equation (2.2) define isothermal coordinates. This, however, follows by direct computation using Proposition 1.5. Examples of minimal surfaces are the surfaces of Sherk (Fig. 2.1) and Catalan (Fig. 2.2). FIGURE 2.1. Sherk s minimal surface. (f g) ( 2 1 z 4 z) An interesting observation is the fact that the Gauss map of minimal surfaces generated using Theorem 2.6 can be identified with the complex valued function g. PROPOSITION 2.7. LetËbe a minimal surface inê3 given by the Weierstrass representation X (z) Re z f (z)(1 g(z) ) 2g(z))dz 2 i(1 g(z) 2 Then the Gauss map N :Ë S projection, with g. z 0 2 ofëmay be identified, via stereographic
20 12 2. MINIMAL SURFACES G 1 FIGURE 2.2. Catalan s minimal surface. (f g) 3 z 1 i z z PROOF. Let 1Æg then Re g 2Im g g G (2 2 1) 1 g2 Now the real and imaginary parts of f (1 g 2 ) if (1 g ) 2fg 2 represent two orthogonal tangent vectors onëinê3. Then fg g i 2 1 g 2 (1 g g) 1 G fg 2 g g g g g and since G 1it is clear that G is a Gauss map forë. We can conclude that the above examples (Figures 2.1, 2.2) have bijective Gauss map i.e. for every point p¾s2 there exists only one point onëhaving that point as a normal. Amongst the Enneper surfaces, defined by (f g) (1 zn ) for every n¾æ, only the case of n 1 satisfies this property (see Figures 2.3, 2.4 and 2.5). FIGURE 2.3. First order Enneper surface. (f g) (1 z)
21 2.2. THE WEIERSTRASS REPRESENTATION FORMULA 13 FIGURE 2.4. Second order Enneper surface. (f g) (1 z 2 ) FIGURE 2.5. Third order Enneper surface. (f g) (1 z 3 )
22
23 CHAPTER 3 CMC surfaces of revolution In this chapter we study complete CMC surfaces with rotational symmetry. We present Kenmotsu s modern solution given in [11] to the problem of finding all such surfaces. Furthermore we describe the classical construction of the same due to Delaunay, see [1]. DEFINITION 3.1. A surfaceëinê3 is said to have constant mean curvature (CMC) if and only if there exists a c¾êsuch that H c Kenmotsu s solution Let : with (s) I Ê Ê2 x(s) y(s) be a parametrization of some regular planar C 2 curve. Assume that is an arclength parametrization and that 0 is contained in the open interval I. LetËbe the surface of revolution inê3 defined by (s ) (x(s) y(s) cos y(s) sin ) s¾i 0 2Ô Then the first and second p (x¼¼y¼ d 2 x¼ fundamental x¼¼yy¼ x¼yy¼¼ 0 2 x¼yd 2 forms are given by I p ds 2 y 2 II x¼y¼¼)ds x¼x¼¼ y¼y¼¼ 0 s¾i s¾i Assuming y(s) 0for s¾i we have, by definition of H, that (3.1) Multiplying by x¼and y¼, respectively, and simplifying using the fact that (x¼) 2 (y¼) 2 1 and we obtain 2Hyx¼ (yy¼)¼ 1 0 and 2Hyy¼ (yx¼)¼ 0 Z¼ 1 0 s¾i Setting Z(s) y(s)y¼(s) iy(s)x¼(s) and combining these equations the following first order complex linear differential equation is obtained 2iHZ (3.2) Restricting our attention to the case of H being constant we have: If H 0 then the solution is given by Z(s) s C s c 1 ic 2 15
24 y c CMC SURFACES OF REVOLUTION for some C c 1 ic 2¾. 2 This gives us y(s) Z(s) Õ(s c 1 ) 2 c2 Z 2 (3.3) x¼(s) Im Ô(s c 1 ) 2 c2 2 By integrating x we obtain 1 x c 2 arcsinh s c s c 1 sinh x c 2 hence c 2 c 2 Substituting into equation (3.3) we obtain y Õ(s c 1 ) 2 c 2 sinh 2 2 x c 2 c 2 c 2 2 c 2 2 cosh x c It is clear that this is a parametrization of a catenary. If H 0 then Z(s) 1 1 e 2iHs C e 2iHs 2iH 1 (1 2iHC) e 2iHs e 2iHs (3.4) 2iH Be i(2hs ) 1 2iH where Be i 1 2iHC for y some B ¾Êand C¾ is an arbitrary constant. Using the fact that y(s) 0 we have by translation of the arclength and by restricting our attention to H 0 y(s) Z 1 2HÔ1 B2 2B sin 2Hs Z 1 B sin 2Hs x¼(s) Im Ô1 B dt 2 2B sin 2Hs Hence the solution to equation (3.4) is the oneparameter family of surfaces of revolution having constant mean curvature H given by 1 Bsin 2Ht Ô1 B2 2B sin 2Ht (s; H B) s 0 1 2HÔ1 B 2 2B sin 2Hs (3.5) for any B¾Êand H 0. Studying for varying B (see Fig. 3.1) we find that (s; H 0) is a generating curve for a right circular cylinder and (s; H 1) is a generating curve for a sequence of continuous halfcircles centered on the xaxis. For 0 B 1 the function x(s) increases monotonously whereas in the case of B 1 it does not.
25 3.2. DELAUNAY S CONSTRUCTION 17 THEOREM 3.2 (Delaunay s theorem). Any complete surface of revolution with constant mean curvature is either a sphere, a catenoid or a surface whose generating curve is given by (s; H B) for some B¾Ê. PROOF. Let H¾Êbe given and let H(s) be a generating curve parametrized by arclength for a complete surface of revolution having constant mean curvature H. By uniqueness of solution of (3.2) we have H(s) (s; H B) for some B¾Ê. FIGURE 3.1. Solutions for H 0 5 and B Delaunay s construction The surfaces of Delaunay are constructed by rolling a conic along a straight line in the plane and taking the trace of the focus F. This is called a roulette of the conic. This trace then describes a planar curve which is rotated about the axis along which it was rolled. This gives a surface of revolution having constant mean curvature. The construction presented here is based on the article [12] by J. Eells is a parabola. Let be a parabola given by : t (t at 2 ) for some a which we take to be strictly positive. Let F be the focus and A be the vertex of (see Figure 3.2). Let K be a point on and denote by P the intersection of the tangent line of at K with the horizontal axis. By solving the line equation for the tangent at K we find that for K (t) (t at 2 ) then P (t 2 0). This implies that PK OP. And since FOP PKF we F K A P O FIGURE 3.2. is parabola
26 18 3. CMC SURFACES OF REVOLUTION FIGURE 3.3. Catenary also. By definition of the trigonometric functions have OPF KPF Ô2 we have FA FP cos AFP FP cos PFK Now let FP denote the xaxis along which our parabola rolls. Then the ordinate of F in this system of coordinates is given by PF. Denote this by y. We have cos PFK dx ds where s is the arclength of the locus of F. This is equivalent to (3.6) ds cos Õ dx where denotes the angle made by the tangent of F with the xaxis. We then arrive at c y dx dx y ds y dx dy ds 2 ds 2 2 Õ1 dy dx 2 or, equivalently, dy c 2 (3.7) c dx Öy 2 The solution to this differential equation is given by y c e x c e x c c cosh x 2 c which is a catenary (Fig. 3.3). The corresponding surface of revolution is the catenoid (Fig. 3.4). The Gauss map of the locus of F into S 1 is given by x x where cos x dx ds c y showing that the Gauss map is injective onto an open semicircle.
27 3.2. DELAUNAY S CONSTRUCTION 19 FIGURE 3.4. Catenoid is an ellipse. Let F and F¼be foci of and O its center. Take a point K on and let P and P¼be the points on the tangent at K closest to F and F¼, respectively (Fig. 3.5). As above, letting PK be the xaxis and PF (P¼F¼) the ordinate y (y¼). Let T and T¼denote the intersection with the xaxis y¼ of the tangent of the locus of F and F¼, respectively. We have FKP F¼KP¼. Also the tangent of the locus of F (F¼) is orthogonal to FK (FL¼) and. This gives us hence KFT KF¼T¼ Ô2 y FK sin FKP cos FTP dx ds F¼K sin F¼KP¼ cos F¼T¼P¼ dx ds 2 From the characterization of the ellipse, FK F¼K 2a for some a 0, and the pedal equation, PF P¼F¼ b for some b 0, we find y y¼ 2a dx and yy¼ b 2 so ds T P K P F O F FIGURE 3.5. is ellipse
28 20 3. CMC SURFACES OF REVOLUTION 2 0 FIGURE 3.6. Undulary, H 0 5, B y 2 2ay dx ds b Taking a b we get the following cases (when the angle is obtuse and acute, respectively) dt dx y 2 2ay (3.8) ds b A solution to this problem is given in Section B sin 2Ht x(s) s 0Ô1 B2 2B sin 2Ht (3.9) y(s) 1 2HÔ1 B2 2B sin 2Hs b where H 1 2. The locus of either foci is called 4H 2 the undulary (Fig. 3.6). The corresponding surfaces is called the unduloid (Fig. 3.7). The Gauss map of the undulary is given by where x x cos x dx 2ay ds y2 b2 2a and B Õ1 FIGURE 3.7. Unduloid, H 0 5, B 0 5.
29 PF P¼F¼ is a hyperbola. We proceed as in the case of the ellipse but instead use the characterization FK F¼K 2a 0 of the hyperbola and the pedal equation b equation 3.2. DELAUNAY S CONSTRUCTION (Fig. 3.8). We arrive at the differential dx y 2 2ay b ds K P F O F P FIGURE 3.8. Hyperbola 2 This differential equation can be solved in the same manner as for the ellipse with the exception that B in equation (3.9) is given by B Ö1 b2 4H Here the two loci fit together to form the curve known as the nodary (Fig. 3.9) and the corresponding surface is called the nodoid (Fig. 3.10) The Gauss map of the nodary is given by where x x cos x y2 b 2ay 2 This map has no extreme points and is clearly surjective. FIGURE 3.9. Nodary, H 0 5, B 1 5.
30 22 3. CMC SURFACES OF REVOLUTION FIGURE Nodoid, H 0 5, B 1 5.
31 CHAPTER 4 CMC surfaces M The main aim of this chapter is to give a new elementary proof of a special case of the RuhVilms theorem. We also present the Kenmotsu representation formula for CMC surfaces with H Harmonicity of the Gauss map Let (M g) be an orientable mdimensional Riemannian manifold, i : ) möh Êm p be an isometric immersion and N : M G p o(êm p ) be the associated Gauss map, mapping x¾mto the oriented normal space of i(m) at i(x). Then RuhVilms theorem presented in [13] states that the tension fieldø(n ) of N satisfies Ø(N whereöh is the covariant derivative of the mean curvature vector field H. This implies that the Gauss map N is harmonic if and only if the mean curvature vector field H is parallel. For surfaces inê3 this is equivalent to the ) surface having constant mean curvature. THEOREM 4.1. LetËbe an oriented surface inê3. ThenËhas constant mean curvature if and only if its Gauss map N :Ë S2 is harmonic. PROOF. We prove that the following equation Ø(N 2 grad H holds. Our sought result then follows trivially. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be isothermal coordinates around p. Then we have y (N xx ) T N xx X x X x ÅN xx X y«x y Ð2 1Ð2 ( e x N x X xx )X x ( f x ÅN y x X yx«)x 1Ð2 ( e x N x X xx )X x ( e y (e g) 2 22 xx ÅN x X yx«)x This follows by using yx«e x x N x X x N xx X x N x X f x xån x X y«ån xx X y«ån x X 23
32 24 4. CMC SURFACES y and the MainardiCodazzi equations (1.10). Similarly we have (N yy ) T 1Ð2 ( g x (e g) 1 11 ÅN y X xy«)x y ( g y ÅN y X yy«)x By using the Christoffel relations (1.9) we 11 y«find N x X xx Åa 11 X x a 21 X y 1 12 X x 2 11 X Ð2 a a 21 2 e 1 11 f 2 ÅN x X 21 yx«ð2 a a 21 2 f 1 11 e 2 ÅN y X yx«ð2 a a 22 2 g 1 11 f 2 ÅN y X 22 x«22 yy«ð2 a a 22 2 f 1 11 g 2 Adding and using the above equations we get 22 N X x ÅN xx N yy X 1Ð2 (e x g x ) (e g) 1 N 11 x X xx xy«y«11 ÅN y X 1Ð2 (e x g x ) 2(e g) 1 22 Å N X ) y«ån xx N T yy X 1Ð2 (e y g y ) 2(e g) 2 Hence we have 1Ð2 Ø(N ( N 1Ð2 e x g x e g ) x X Ð2 x 1Ð2 e y g y e g y X Ð2 y 1Ð2 e x g x 1Ð2 ) X x 1Ð2 e y g y 1Ð2 H ) X y 1Ðx (e g) 1Ð2 (e g) 1Ðx ( 1Ð2 ) 1ÐX x 1Ðy (e g) 1Ð2 (e g) 1Ðy ( 1Ð2 ) 1ÐX y 2 grad This proves our theorem Kenmotsu s representation formula Ð4Ð2 Ð4Ð2 Ð2 (e g)x ( Ð2 (e g)y ( In this section we show a corresponding result to Weierstrass representation formula for surfaces having nonzero constant mean curvature.
33 Let S 2 be the unit sphere inê3. Cover S 2 by open sets U i, i 1 2 where U 1 S2 n and U 2 S2 s and n and s are the north and south pole, respectively. Let be the stereographic projection with respect 1 to the north pole n: (x) x 1 ix 2 2 for x (x 1 x 2 x 3 )¾U (4.1) 1 3 For a surfaceëinê3 having Gauss map N :Ë S2 consider the following composition :ËN S which we also call the Gauss map ofë. This map is considered as a complex mapping from a 1dimensional complex manifoldëinê3 into the Riemann sphere. Using this notation we have the following theorem presented in [14] due to K. Kenmotsu KENMOTSU S REPRESENTATION FORMULA 25 THEOREM 4.2 (Kenmotsu s representation formula). Let V be an open simply connected subset of and H be an arbitrary nonzero real constant. Suppose : V is a harmonic function into the Riemann sphere. If z 0 then X : V Ê3 defined by X (z) Re z X z z 0 (z)dz (4.2) with ( 1) X z (z) ) 2 (z) (z) 1 (z) 2 i(1 (z) 2 H(1 (z) (z)) 2 z for z¾v, is a regular surface having as a Gauss map and mean curvature H. Ø( ) 4Ð z z First we derive an explicit formula of the tension field of the Gauss map :Ë of an arbitrary surfaceë. PROPOSITION 4.3. LetËbe a surface inê3 and :Ë be a Gauss map onë. If z x iy are local isothermal coordinates with dilationðthen (4.3) z z PROOF. Let z x iy be local isothermal coordinates and denote (z) u iv. Then 4 (1 u2 v2 ) (du2 dv 2 ) 2 which gives us Christoffel symbols 12 on the Riemann sphere u u 2 v 2v u 2 v h g Ð2 (dx 2 dy 2 ) and
34 y CMC SURFACES 11 u x 2 u Ø(u) 1Ð2 u 1 22 v x 2 v 1 By the explicit formula for the tension field (eq. A.2) we get 12 u xv y y x u yv x 2 y 2 v x v y Ø(u) iø(v) where is the classical Laplacian. Adding this and the similar formula forø(v) we arrive at Ð2Ø( ) Ð2 (u iv) u iv 2 u x 2 2 2iu xv u 1 2 v z z u y 2 2iu yv 8 z 8 z z z This proves our sought formula. H z Ý z :Ë Ê Next we show that a surface having prescribed mean curvature H satisfies the following equation. LEMMA 4.4. LetËbe a surface inê3 having mean curvature H and let :Ë be a Gauss map ofë. Then z whereý 1Ð2 (e g) 2 if. PROOF. We show that the following equation holds H 1 2(X 1 ix 2 ) 3 2 x 2 By direct computation using the Weingarten equations (1.11) together with equation (4.1) we have z z N 1 in N 2Ð2 (1 N 3 ) Xx N 1 3 Xx 1 N 1 X 3 i Xx 2 N 3 Xx N 2 2 Xx e 3 z (4.4) (1 N 3 ) X 2 y X 2 x ixy ix 2 x (N 1 1 in 2 ) Xy ix 3 x f 3
Mathematics 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 informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
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 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 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 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 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 informationON DISCRETE CONSTANT MEAN CURVATURE SURFACES
ON DISCRETE CONSTANT MEAN CURVATURE SURFACES CHRISTIAN MÜLLER Abstract. Recently a curvature theory for polyhedral surfaces has been established which associates with each face a mean curvature value computed
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 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 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 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 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 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 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 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 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 informationEuler  Savary s Formula on Minkowski Geometry
Euler  Saary s Formula on Minkowski Geometry T. Ikawa Dedicated to the Memory of Grigorios TSAGAS (935003) President of Balkan Society of Geometers (997003) Abstract We consider a base cure a rolling
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 informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the CauchyRiemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationMATH 381 HOMEWORK 2 SOLUTIONS
MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e
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 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 informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
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 information14.11. 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 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 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 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 informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
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 informationWe can display an object on a monitor screen in three different computermodel forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
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 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 informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
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 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 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 informationAPPLICATIONS OF TENSOR ANALYSIS
APPLICATIONS OF TENSOR ANALYSIS (formerly titled: Applications of the Absolute Differential Calculus) by A J McCONNELL Dover Publications, Inc, Neiv York CONTENTS PART I ALGEBRAIC PRELIMINARIES/ CHAPTER
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 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 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 informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More information16.5: CURL AND DIVERGENCE
16.5: URL AN IVERGENE KIAM HEONG KWA 1. url Let F = P i + Qj + Rk be a vector field on a solid region R 3. If all firstorder partial derivatives of P, Q, and R exist, then the curl of F on is the vector
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
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 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 informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
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 informationParametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) YanBin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
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 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 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 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 informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
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 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 informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
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 informationNotes on the representational possibilities of projective quadrics in four dimensions
bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and
More information7. Cauchy s integral theorem and Cauchy s integral formula
7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationDouble Tangent Circles and Focal Properties of SpheroConics
Journal for Geometry and Graphics Volume 12 (2008), No. 2, 161 169. Double Tangent Circles and Focal Properties of SpheroConics HansPeter Schröcker Unit Geometry and CAD, University Innsbruck Technikerstraße
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 informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
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 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 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 informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
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 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 informationOn Motion of Robot EndEffector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot EndEffector using the Curvature
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 information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
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 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 informationLecture 17: Conformal Invariance
Lecture 17: Conformal Invariance Scribe: Yee Lok Wong Department of Mathematics, MIT November 7, 006 1 Eventual Hitting Probability In previous lectures, we studied the following PDE for ρ(x, t x 0 ) that
More informationSOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS
SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the rth
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
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 information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More information1. Firstorder Ordinary Differential Equations
Advanced Engineering Mathematics 1. Firstorder ODEs 1 1. Firstorder Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
More information4B. Line Integrals in the Plane
4. Line Integrals in the Plane 4A. Plane Vector Fields 4A1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region
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 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 informationFIXED POINT SETS OF FIBERPRESERVING MAPS
FIXED POINT SETS OF FIBERPRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 email: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics
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 informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
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 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 informationLine and surface integrals: Solutions
hapter 5 Line and surface integrals: olutions Example 5.1 Find the work done by the force F(x, y) x 2 i xyj in moving a particle along the curve which runs from (1, ) to (, 1) along the unit circle and
More information