Mean Curvature Flow in Higher Codimension


 Sophie Robinson
 1 years ago
 Views:
Transcription
1 Mean Curvature Flow in Higher Codimension Summer School on Differential Geometry Konkuk University, Seoul August 2013 Knut Smoczyk Leibniz Universität Hannover
2
3 Introduction Mean curvature flow is perhaps the most important geometric evolution equation of submanifolds in Riemannian manifolds. Intuitively, a family of smooth submanifolds evolves under mean curvature flow, if the velocity at each point of the submanifold is given by the mean curvature vector at that point. For example, round spheres in euclidean space evolve under mean curvature flow while concentrically shrinking inward until they collapse in finite time to a single point, the common center of the spheres. Mullins [Mul56] proposed mean curvature flow to model the formation of grain boundaries in annealing metals. Later the evolution of submanifolds by their mean curvature has been studied by Brakke [Bra78] from the viewpoint of geometric measure theory. Among the first authors who studied the corresponding nonparametric problem were Temam [Tem76] in the late 1970 s and Gerhardt [Ger80] and Ecker [Eck82] in the early 1980 s. Pioneering work was done by Gage [Gag84], Gage & Hamilton [GH86] and Grayson [Gra87] who proved that the curve shortening flow (more precisely, the mean" curvature flow of curves in R 2 ) shrinks embedded closed curves to round" points. In his seminal paper Huisken [Hui84] proved that closed convex hypersurfaces in euclidean space R m+1,m > 1 contract to single round points in finite time (later he extended his result to hypersurfaces in Riemannian manifolds that satisfy a suitable stronger convexity, see [Hui86]). Then, until the mid 1990 s, most authors who studied mean curvature flow mainly considered hypersurfaces, both in euclidean and Riemannian manifolds, whereas mean curvature flow in higher codimension did not play a great role. There are various reasons for this, one of them is certainly the much different geometric situation of submanifolds in higher codimension since the normal bundle and the second fundamental tensor are more complicated. But also the analysis becomes more involved and the algebra of the second fundamental tensor is much more subtle since for hypersurfaces there usually exist more scalar quantities related to the second fundamental form than in case of submanifolds in higher codimension. Some of the results previously obtained for mean curvature flow of hypersurfaces carry over without change to submanifolds of higher codimension but many do not and in addition even new phenomena occur. Among the first results in this direction are the results on mean curvature flow of space curves by Altschuler and Grayson [Alt91,AG92], measuretheoretic approaches to higher codimension mean curvature flows by Ambrosio & Soner [AS97], existence and convergence results for the Lagrangian mean curvature flow [Smo96, Smo00, Smo02,TY02], mean curvature flow of symplectic surfaces in codimension two [CL04, Wan02] and longtime existence and convergence results of graphic mean curvature flows in higher codimension [CLT02, SW02, Smo04, Wan02, Xin08]. Recently there i
4 has been done quite some work on the formation and classification of singularities in mean curvature flow [Anc06, CL10, CCH09a, CSS07, CM09, GSSZ07, HL09, HS09, JLT10, LS10a, LS10b, LXYZ11, SW03], partially motivated by Hamilton s and Perelman s [Ham95a, Per02, Per03a, Per03b] work on the Ricci flow that in many ways behaves akin to the mean curvature flow and vice versa. The results in mean curvature flow can be roughly grouped into two categories: The first category contains results that hold (more or less) in general, i.e. that are independent of dimension, codimension or the ambient space. In the second class we find results that are adapted to more specific geometric situations, like results for hypersurfaces, Lagrangian or symplectic submanifolds, graphs, etc.. These notes are based on my lectures on mean curvature flow in higher codimension that I held during the Summer School on Differential Geometry at Konkuk University in Seoul, August My aim in these lectures was to give an introduction to mean curvature flow for the beginner. For those interested in a more detailed description, let me refer to several nice monographs on mean curvature flow that can be found in the literature, e.g. a well written introduction to the regularity of mean curvature flow of hypersurfaces is [Eck04]. For the curve shortening flow see [CZ01]. For mean curvature flow in higher codimension there exist some lecture notes by Wang [Wan08b] and a detailed introduction to higher codimensional mean curvature flow by myself [Smo12]. Konkuk University in Seoul, August 2013 Knut Smoczyk ii
5 Contents Lecture Immersions and embeddings Tangent and normal bundles st fundamental form Connections nd fundamental form and mean curvature vector Lecture Structure equations Local coordinates First variation of volume Mean curvature flow Invariance under ambient isometries Lecture Invariance under the diffeomorphism group Analytic nature of the mean curvature flow Shorttime existence and uniqueness Longtime existence Evolution equations Lecture Maximum principle Comparison principles Lagrangian submanifolds Graphs Selfsimilar solutions Bibliography 27 Index 33 iii
6
7 Lecture 1 Throughout these lectures, M will be an oriented smooth manifold of dimension m and (N,g) will be a complete (either compact or noncompact) smooth Riemannian manifold of dimension n > m. 1 Immersions and embeddings For 1 k we will consider C k Immersions of M into N, i.e. maps F : M N of class C k for which the Differential DF p : T p M T F(p) N is injective for any p M. An immersion F : M N will be called an embedding, if F(M) N is an embedded submanifold. a) b) Figure 2.1: a) An immersion of M = S 1 into N = R 2. b) Embedding of S 1 into R 2. 2 Tangent and normal bundles If F : M N is an immersion and p M we define T p M := { w T F(p) N : v T p M with DF p (v) = w } 1
8 Lecture 1 and T p M := { w T F(p) N : g F(p) (w, w) = 0, w T p M }, i.e. T p M is the orthogonal complement w.r.t. g F(p) of T p M within T F(p) N. The normal bundle of M is the bundle T M over M whose fibers are given by T p M. Analogously we define the bundle T M as the bundle over M with fibers T p M. Both bundles are subbundles of the tangent bundle of N along M, i.e. of the bundle F T N over M whose fiber at p M is given by T F(p) N. Sometimes F T N is called the pullback bundle of T N via F. Since a) b) Figure 2.2: a) T p M can be identified with T p M but is not the same vector space. b) The two projections of w T q N onto w T p M and w T p M with q = F(p). DF p : T p M T p M is an isomorphism for each p M, one usually identifies T M with T M. We obtain two natural projections : (F T N) p T p M w w : (F T N) p T p M w w and for any p F 1 (q) we obtain the orthogonal decomposition T q N = T p M T p M = DF p ( Tp M ) T p M. 3 1st fundamental form 3.1 Definition (Induced Riemannian metric (1st fundamental form)) If F : M N is an immersion into a Riemannian manifold (N,g), then F induces a Riemannian metric F g on M, via (F g) p (v 1,v 2 ) := g F(p) (DF p (v 1 ),DF p (v 2 )), p M and v 1,v 2 T p M. 2
9 4 Connections It is important that F is an immersion since otherwise F g might be degenerated. 4 Connections The connections we use are all induced by the Riemannian metric g and its Levi Civita connection T N on T N. We recall that the LeviCivita connection on T N is defined by the formula g( T V N X,Y ) = 1 (V ( g(x,y ) ) + X ( g(v,y ) ) Y ( g(v,x) ) 2 ) g(v,[x,y ]) g(x,[v,y ]) + g(y,[v,x]), where V,X,Y X(N) are arbitrary smooth vector fields. Let F : M N be a smooth immersion. On F T N we use the pullback connection F T N. This can be described as follows. Let W Γ (F T N) be a smooth section in the pullback bundle, i.e. let W : M F T N be a smooth map with W (p) T F(p) N for all p M. Let p M be arbitrary and set q := F(p). In an open neighborhood V N of q we choose a local trivialization of T N, e.g. (e α ) α=1,...,n, e α X(T V ). If U M is a sufficiently small open set containing p, then W U can be written in the form W (p ) = W α (p )e α (F(p )), p U with smooth functions W α C (U). For v T p M we then set ( F T N v W ) (p) := DW α p (v)e α(f(p)) + W α (p) ( ) T v N e α (F(p)). This is welldefined, i.e. independent of the choice of the frame (e α ) α=1,...,n. Since T M and T M are both subbundles of F T N we obtain connections on them by v W := ( F T N v W ), v ν := ( F T N v ν ), where v T p M and W Γ (T M) and ν Γ (T M) are smooth sections. Note that by definition of the LeviCivita connection w.r.t. the induced metric F g on T M and by definition of the pullback connection and bundle we have for any X X(M) that DF(X) Γ (T M) and (4.1) DF ( v X ) ( ) = v DF(X). 3
10 Lecture 1 Since in general it is clear how the connections are induced on all bundles over M that naturally appear in our context, we will sometimes omit the superscript, i.e. we will then simply use for all these connections. We make an exception for the connection on the normal bundle and on product bundles E T M containig T M as a factor. 5 2nd fundamental form and mean curvature vector If F : M N is an immersion, then the differential DF maps T p M to T M T F(p) N and hence DF Γ (T M T M) can also be considered as a section DF Γ (F T N T M). We extend the projections, to product bundles containing either T M or T M as a factor. In this way we have for any vector field X X(M) and ( T M T M v DF ) (X) = v (DF(X)) DF( v X) (4.1) = 0 (5.1) ( F T N T M v DF ) (X) = F T N v (DF(X)) DF( v X) = ( F T N v (DF(X)) ) ( + F T N v (DF(X)) ) DF( v X) = ( F T N v (DF(X)) ) + v (DF(X)) DF( v X) = ( F T N v (DF(X)) ). 2nd fundamental tensor The second fundamental tensor A of an immersion F : M N is given by A := F T N T M DF Γ (F T N T M T M), A p (v 1,v 2 ) = ( F T N T M v 1 DF ) p (v 2) for all v 1,v 2 T p M. 5.1 Lemma Let A be the second fundamental tensor of an immersion F : M N. a) A is normal, i.e. g F(p) ( A p (v 1,v 2 ),DF p (v 3 ) ) = 0, v 1,v 2,v 3 T p M. Consequently, A can also be considered as a section in T M T M T M. b) A is symmetric, i.e. A p (v 1,v 2 ) = A p (v 2,v 1 ), v 1,v 2 T p M and p M. 4
11 = ( DF([V 1,V 2 ]) ) = 0 5 2nd fundamental form and mean curvature vector Proof: a) Directly from equation (5.1). b) Let V 1,V 2 X(M) be smooth vector fields. Since T N and are torsion free we get Hence F T N V 1 (DF(V 2 )) F T N V 2 (DF(V 1 )) = DF([V 1,V 2 ]). A(V 1,V 2 ) A(V 2,V 1 ) = ( F T N V 1 (DF(V 2 )) ) ( F T N V 2 (DF(V 1 )) ) 2nd fundamental form If ν T p M is a normal vector, then the second fundamental form A ν w.r.t. ν is the symmetric bilinear form on T p M defined by A ν (v 1,v 2 ) := g F(p) (A p (v 1,v 2 ),ν), for all v 1,v 2 T p M. Mean curvature vector The mean curvature vector field H of an immersion F : M N is the trace of the second fundamental tensor A. At p M the vecor H p is therefore given by H p = m A p (e k,e k ), k=1 where (e k ) k=1,...,m is an arbitrary orthonormal basis of T p M. Since A is normal this holds for H as well, i.e. H Γ (T M) is a smooth normal vector field. 5
12
13 Lecture 2 6 Structure equations A connection on a vector bundle E over a manifold induces a curvature tensor R(V,W )Φ = V W Φ W V Φ [V,W ] Φ, where Φ Γ (E) and V,W are smooth vector fields on that manifold. Since R(V,W )Φ is C linear in each argument, i.e. f R(V,W )Φ = R(f V,W )Φ = R(V,f W )Φ = R(V,W )(f Φ), for all smooth f, one gets that ( R(V,W )Φ ) depends only on v = V (p),w = W (p) and ϕ = Φ(p). This p means that at p the quantity R(v,w)ϕ is welldefined (just extend v,w,ϕ to smooth sections V,W,Φ and compute ( R(V,W )Φ ) ). p The structure equations of an immersion F : M N of a smooth manifold M into a Riemannian manifold (N,g) give relations between the curvature tensors R M, R N, R and the second fundament form A of F. Here, R M is the Riemannian curvature tensor of T M w.r.t. the LeviCivita connection and R N is the curvature tensor of T N w.r.t. T N. Moreover, R denotes the curvature tensor of the normal bundle T M w.r.t.. Gauß equations R M (v 1,w 1,v 2,w 2 ) R ( N DF p (v 1 ),DF p (w 1 ),DF p (v 2 ),DF p (w 2 ) ) n m ( ( = g F(p) A ν k (v 1,v 2 ),A ν k (w 1,w 2 ) ) ( g F(p) A ν k (v 1,w 2 ),A ν k (w 1,v 2 ) )), k=1 where (ν k ) k=1,...,n m is an orthonormal basis of T p M and v 1,v 2,w 1,w 2 T p M. 7
14 Lecture 2 Ricci equations ( R M (v,w)ν R ( N DF p (v),df p (w) ) ) m ( ) ν = A ν (w,e k )A(v,e k ) A ν (v,e k )A(w,e k ), where (e k ) k=1,...,m is an orthonormal basis of T p M and ν T p M, v,w T p M. k=1 Codazzi equations ( u A)(v,w) ( v A)(u,w) = where ν T p M, u,v,w T p M. ( R N ( DF p (u),df p (v) )( DF p (w) )), 7 Local coordinates For computations one often needs local expressions of tensors. Whenever we use local expressions and F : M N is an immersion we make the following general assumptions and notations. i) (U,x,Ω) and (V,y,Λ) are local coordinate charts around p U M and F(p) V N such that F U : U F(U) is an embedding and such that F(U) V. From the coordinate functions (x i ) i=1,...,m : U Ω R m, (y α ) α=1,...,n : V Λ R n we obtain a local expression for F, y F x 1 : Ω Λ, F α := y α F x 1, α = 1,...,n. ii) The Christoffel symbols of the LeviCivita connections on M resp. N will be denoted by Γjk i, i,j,k = 1,...,m, resp. Γ βγ α, α,β,γ = 1,...,n. iii) All indices referring to M will be denoted by Latin minuscules and those related to N by Greek minuscules. Moreover, we will always use the Einstein convention to sum over repeated indices from 1 to the corresponding dimension. 8
15 7 Local coordinates Figure 3.1: Local description of a smooth map F : M N. Example 1 The local expressions of g,df and F g are g = g αβ dy α dy β, DF = F α i y α dxi, F α i := Fα x i, F g = g ij dx i dx j, g ij := g αβ F α i Fβ j. 9
16 Lecture 2 Example 2 The local expression for the 2nd fundamental tensor A is A = A ij dx i dx j = A α ij y α dxi dx j, where the coefficients A α ij are given by the Gauß formula (7.1) A α ij = 2 F α x i x j Γ ij k F α x k + Γ βγ α F β F γ x i x j. Let (g ij ) denote the inverse matrix of (g ij ) so that g ik g kj = δ i j gives the Kronecker symbol. (g ij ) defines the metric on T M dual to F g. For the mean curvature vector we get (7.2) H = H α y α, Hα := g ij A α ij. 8 First variation of volume Let us assume that F 0 : M N is a smooth immersion of an oriented manifold M into a Riemannian manifold (N,g). Then F 0 induces a volume form µ 0 on M. In local positively oriented coordinates (x i ) i=1,...,m the volume form takes the form µ 0 = det(f 0 g) dx1 dx m. If K M is compact, the volume of K w.r.t. µ 0 is vol 0 (K) := µ 0. K 10
17 8 First variation of volume A compactly supported variation of F 0 is a smooth map F : M ( ϵ,ϵ) N, ϵ > 0, such that (i) F(p,0) = F 0 (p) for all p M. (ii) For each t ( ϵ,ϵ) the map F t : M N, F t (p) := F(p,t) defines a smooth immersion into N. (iii) There exists a compact subset K M such that for each t 0 ( ϵ,ϵ) we have ( ) d supp F dt t K. t 0 For a compactly supported variation let us set ϕ t := d dt F t. Then ϕ t Γ (Ft T N) has compact support for any t ( ϵ,ϵ). We want to compute d dt (F t g). Let V 1,V 2 X(M) be time independent smooth vector field on M. Then ( ) d dt (F t g) (V 1,V 2 ) = d ( ) (8.1) (Ft g)(v dt 1,V 2 ) = d ( ) g(df dt t (V 1 ),DF t (V 2 )) = g ( F t T N V 1 ϕ t,df t (V 2 ) ) + g ( DF t (V 1 ), F t T N ) V 2 ϕ t = g ( F t T N V 1 +g ( F t T N V 1 = g ( ϕt, F t T N V 1 +g ( F t T N V 1 = 2g ( ϕt,a t(v 1,V 2 ) ) +g ( F t T N V 1 ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ϕt ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ) ϕ t ) ( DFt (V 1 ) ),ϕt ) ( DFt (V 2 ) )) g ( F t T N V 2 ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ϕt ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N ) V 2 ϕt. Consequently we derive for the variation of the volume form µ t w.r.t. Ft g d dt µ t = 1 ( ) d 2 trace dt (F t g) µ t = g(ϕ t,h t)µ t + d(τ t µ t ), ) 11
18 Lecture 2 where τ t X(M) is the tangent vector field with DF t (τ t ) = ϕt and H t denotes the mean curvature vector field at time t. Therefore, by Stokes theorem and since ϕ t is compactly supported we obtain d ( volt (K) ) = g(ϕt dt,h t)µ t. From this it follows The L 2 gradient of the volume functional is given by H. F : M N is called a minimal immersion, if H = 0. 9 Mean curvature flow Definition A smooth family of immersions F t : M N, t [0,T ), 0 < T, is called a solution of the mean curvature flow, if F t satisfies the evolution equation K d dt F t = H t, where H t is the mean curvature vector field w.r.t. F t. Figure 3.2: Spheres shrink homothetically to points in finite time Example Round spheres S m (R) R n shrink by a family of round spheres S m (r(t)) centered at the same point with r(t) = R 2 2mt. In particular, in this case the flow exists only on a finite time interval [0,T ) with T = R 2 /2m. 12
19 10 Invariance under ambient isometries 10 Invariance under ambient isometries The mean curvature flow is isotropic, i.e. invariant under isometries of the ambient space. This property follows from the invariance of the first and second fundamental forms under isometries. Invariance under isometries Suppose F : M [0,T ) N is a smooth solution of the mean curvature flow and assume that ϕ is an isometry of the ambient space (N,g). Then the family F := ϕ F is another smooth solution of the mean curvature flow. In particular, if the initial immersion is invariant under ϕ, then it will stay invariant for all t [0,T ). 13
20
21 Lecture 3 11 Invariance under the diffeomorphism group Writing a solution F : M N of H = 0 locally as the graph over its tangent plane at F(p), we see that we need as many height functions as there are codimensions, i.e. we need k = n m functions. On the other hand the system H = 0 consists of n coupled equations and is therefore overdetermined with a redundancy of m equations. These m redundant equations correspond to the diffeomorphism group of the underlying mdimensional manifold M. This fact also applies to the mean curvature flow and implies the following: Invariance under the diffeomorphism group If F : M [0,T ) N is a solution of the mean curvature flow, and ψ Diff(M) a fixed diffeomorphism of M, then F : M [0,T ) N, F(p,t) := F(ψ(p),t) is another solution. In particular, for each t [0,T ) the immersed submanifolds M t := F(M,t) and M t := F(M,t) coincide. 12 Analytic nature of the mean curvature flow Using local coordinates, we may easily get insight into the analytic nature of the mean curvature flow. From Gauß equation we see that locally ( d dt Fα (x,t) = g ij 2 F α (x,t) x i x j (x,t) Γ ij k (x,t) Fα x k (x,t) + Γ ( ) F βγ α β ) F(x,t) x i (x,t) Fγ x j (x,t). Thus the mean curvature flow is a degenerate quasilinear parabolic system of second order, where the k = n m degenerecies stem from the invariance under the diffeomorphism group. Since H = tracea = trace( DF), we may also consider the mean curvature flow as the heat equation 15
22 Lecture 3 Mean curvature flow = heat equation on the space of immersions d dt F t = H t = t F t on the space of smooth immersion of a given manifold M into a Riemannian manifold (N,g), where t denotes the LaplaceBeltrami operator w.r.t. F t g. 13 Shorttime existence and uniqueness The following theorem is wellknown and in particular forms a special case of a theorem by Richard Hamilton [Ham82b], based on the NashMoser implicit function theorem treated in another paper by Hamilton [Ham82a]. Shorttime existence and uniqueness Let M be a smooth closed manifold and F 0 : M N a smooth immersion into a smooth Riemannian manifold (N, g). Then the mean curvature flow admits a unique smooth solution on a some short time interval [0,ϵ), ϵ > 0. Figure 4.1: Embedded closed curves in R 2 shrink to "round" points. 14 Longtime existence In general one does not have longtime existence of a solution. 16
23 15 Evolution equations Example Suppose F 0 : M R n is a smooth immersion of a closed mdimensional manifold M. Then the maximal time T of existence of a smooth solution F : M [0,T ) R n of the mean curvature flow with initial immersion F 0 is finite. The next well known theorem holds in any case. Longtime existence criterion Let M be a closed manifold and F : M [0,T ) (N,g) a smooth solution of the mean curvature flow in a complete (compact or noncompact) Riemannian manifold (N,g). Suppose the maximal time of existence T is finite. Then lim sup t T ( max A 2) =. M t 15 Evolution equations From the main evolution equation d dt F t = H t we obtain the evolution equations of all relevant geometric quantities, e.g. the evolution equation of the induced metric F t g is d dt F t g = 2A H t t = 2g(H t,a t ). This immediately follows from (8.1). Moreover the volume form evolves by d dt µ t = H t 2 µ t, so that the volume is always decreasing, if F t is not a minimal immersion. 17
24
25 Lecture 4 16 Maximum principle One key technique in mean curvature flow are maximum principles. To demonstrate this we will give an example. Example Let F t : M R n, t [0,T ), be a mean curvature flow and suppose M is compact. Then the maximal time of existence T is finite. Proof: We compute the evolution equation of f := F t 2. d dt f = 2 F t, d dt F t = 2 F t,h t = 2 F t, t F t = t F t 2 2 t F t 2 = t f 2m, because t F t 2 = g ij Ft t x i, F t x j } {{ } = dimm = m. =(g t ) ij Therefore the function p := f + 2mt satisfies d dt p = tp. The weak parabolic maximum principle now states that for any t 0 (0,T ) we have Therefore in particular for all x M max p(x,t) max p(x,0). (x,t) M [0,t 0 ] x M p(x,t 0 ) = F t0 (x) 2 + 2mt 0 max x M F 0(x) 2. Since this holds for any t 0 < T we obtain T <. 19
26 Lecture 4 Remark In general one can say the following: If a function f satisfies an evolution equation of the form d dt f = tf + t f,v t + ϕ(f ) for a smooth function ϕ and a smooth vector field V t, then f behaves in the worst case as the solution of the ODE d f = ϕ(f ). dt 17 Comparison principles From the maximum principle one can deduce the following comparison principle. Comparison principle Let M 1,M 2 be mdimensional and let N have dimension n = m + 1. If F i : M i [0,T i ) N, i = 1,2, are two (immersed) mean curvature flows and F 1 (M 1,0) F 2 (M 2,0) =, then this holds for all t [0,min{T 1,T 2 }). i.e. F 1 (M 1,t) F 2 (M 2,t) =, provided at least one of the manifolds M 1,M 2 is compact. In the same way one can prove that embeddedness is preserved, if the codimension is again one. Embeddedness Suppose F : M [0,T ) N is a mean curvature flow of a compact hypersurface and suppose F(M,0) is embedded. Then F(M,t) is embedded for all t [0,T ). 18 Lagrangian submanifolds Let (N,g =,,J) be a Kähler manifold, i.e. J End(T N) is a parallel complex structure compatible with g. Then N becomes a symplectic manifold with the symplectic form ω given by the Kähler form ω(v,w ) = JV,W. An immersion F : M N is called Lagrangian, if F ω = 0 and n = dimn = 2m = 2dimM. For a Lagrangian immersion we define a section ν Γ (T M T M), ν := JDF, 20
27 18 Lagrangian submanifolds where J is applied to the F T Npart of DF. ν is a 1form with values in T M since by the Lagrangian condition J induces a bundle isomorphism (actually even a bundle isometry) between DF(T M) and T M. In local coordinates ν can be written as with Since J is parallel, we have ν = ν i dx i = ν α i y α dxi ν i = JF i = J α β Fβ i y α, να i = Jα β Fβ i. ν = J DF = JA. Second fundamental form We may define a second fundamental form as a trilinear form It turns out that h is fully symmetric. h(x,y,z) := ν(x),a(y,z). Mean curvature form Taking a trace, we obtain a 1form H Ω 1 (M), called the mean curvature form, In local coordinates H(X) := traceh(x,, ). h = h ijk dx i dx j dx k, H = H i dx i, H i = g kl h ikl. The second fundamental tensor A and the mean curvature vector H can be written in the form A α ij = h k ij να k, H = H k ν k. Since J gives an isometry between the normal and tangent bundle of M, the equations of Gauß and Ricci coincide, so that we get the single equation R ijkl = R N (F i,f j,f k,f l ) + h ikm h m jl h ilm h m jk. 21
28 Lecture 4 Since J = 0 and J 2 = Id we also get i ν α j = i(j α β Fβ j ) = Jα β if β j = Jα β Aβ ij = Jα β νβ k h k ij Similarly as above we conclude i h jkl j h ikl = i A jk,ν l j A ik,ν l Taking a trace over k and l, we deduce ( ν l DF(T M)) = i A jk j A ik,ν l = R N (ν l,f k,f i,f j ). i H j j H i = R N (ν k,f k,f i,f j ) = h k ij Fα k. and if we take into account that N is Kähler and M Lagrangian, then the RHS is a Ricci curvature, so that the exterior derivative dh of the mean curvature form H is given by (dh) ij = i H j j H i = Ric N (ν i,f j ). Lagrangian angle If (N,g,J) is KählerEinstein, then H is closed (since Ric N (ν i,f j ) = c ω(f i,f j ) = 0) and defines a cohomology class on M. In this case any (in general only locally defined) function α with dα = H is called a Lagrangian angle. In some sense the Lagrangian condition is an integrability condition. If we represent a Lagrangian submanifold locally as the graph over its tangent space, then the m height" functions are not completely independent but are related to a common potential. An easy way to see this, is to consider a locally defined 1form λ on M (in a neighborhood of some point of F(M)) with dλ = ω. Then by the Lagrangian condition 0 = F ω = F dλ = df λ. So F λ is closed and by Poincaré s Lemma locally integrable. By the implicit function theorem this potential for λ is related to the height functions of M (cf. [Smo00]). Note also that by a result of Weinstein for any Lagrangian embedding M N there exists a tubular neighborhood of M which is symplectomorphic to T M with its canonical symplectic structure ω = dλ induced by the Liouville form λ. 22
29 19 Graphs 19 Graphs Graphs Let (M,g M ), (K,g K ) be two Riemannian manifolds and f : M K a smooth map. f induces a graph Γ f := F(M) M K, where F : M N := M K, F(p) := (p,f (p)). Since N is also a Riemannian manifold equipped with the product metric g = g M g K one may consider the geometry of such graphs. It is clear that the geometry of F must be completely determined by f, g M and g K. Local coordinates (x i ) i=1,...,m, (z A ) A=1,...,k for M resp. K induce local coordinates (y α ) α=1,...,n=m+k on N by y = (x,z). Then locally F i (x) = x i + f A i (x) z A, where similarly as before f A = z A f x 1 and f A i = f A x i. First fundamental form of graphs For the induced metric F g = g ij dx i dx j we get F g = g M + f g K. Since this is obviously positive definite and F is injective, graphs F : M M K of smooth mappings f : M K are always embeddings. From the formula for DF = F i dx i and the Gauß formula one may then compute the second fundamental tensor A = DF. PseudoRiemannian metric The tensor s = g M ( g K ) defines a pseudoriemannian metric on the product manifold. The tensor F s = g M f g K turns out to be very important in the analysis of the flow. The eigenvalues µ k of F s w.r.t. F g are given by where λ k are the singular values of f. µ k = 1 λ2 k 1 + λ 2 k 23
30 Lecture 4 Length decreasing maps (contractions) A map f : M K is called length decreasing, if Equivalently F s 0. f g K g M. Area decreasing A map f : M K is called area decreasing, if df (v) df (w) g K v w g M, v,w T M. Equivalently, F s is twopositive, i.e. µ k + µ l 0 for all k l. Isotopies Under certain conditions on the curvatures of M, K one can prove that area decreasing (or length decreasing) maps can be deformed into constant maps through mean curvature flow isotopies. 20 Selfsimilar solutions Let F : M R n be an immersion. Selfshrinker F is called a selfshrinker, if H = F. Selfexpander F is called a selfexpander, if H = F. Translator F is called a translator, if for some fixed nonzero vector V R n. H = V, 24
31 20 Selfsimilar solutions These submanifolds appear as special solutions of the mean curvature flow and they serve as models for certain singularities. Example Any minimal immersion F : M S n is a selfshrinker in R n+1. 25
32
33 Bibliography [AL86] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), no. 2, [Alt91] St. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom. 34 (1991), no. 2, [AG92] St. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, J. Differential Geom. 35 (1992), no. 2, [AS97] L. Ambrosio and H. M. Soner, A measuretheoretic approach to higher codimension mean curvature flows, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 12, (1998). Dedicated to Ennio De Giorgi. [AB10] B. Andrews and C. Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010), no. 3, [Anc06] H. Anciaux, Construction of Lagrangian selfsimilar solutions to the mean curvature flow in C n, Geom. Dedicata 120 (2006), [Ang91] S. Angenent, On the formation of singularities in the curve shortening flow, J. Differential Geom. 33 (1991), no. 3, [AV97] S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math. 482 (1997), [Beh08] T. Behrndt, Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein, arxiv: , to appear in Proceedings of CDG 2009, Leibniz Universität Hannover (2008). [Bra78] K. A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., [CL10] I. Castro and A. M. Lerma, Hamiltonian stationary selfsimilar solutions for Lagrangian mean curvature flow in the complex Euclidean plane, Proc. Amer. Math. Soc. 138 (2010), no. 5, [CCH09a] A. Chau, J. Chen, and W. He, Entire selfsimilar solutions to Lagrangian Mean curvature flow, arxiv: (2009). [CCH09b], Lagrangian Mean Curvature flow for entire Lipschitz graphs, arxiv: (2009). [CGG91] Y. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, [CJL05] X. Chen, H. Jian, and Q. Liu, Convexity and symmetry of translating solitons in mean curvature flows, Chinese Ann. Math. Ser. B 26 (2005), no. 3, [CL01] J. Chen and J. Li, Mean curvature flow of surface in 4manifolds, Adv. Math. 163 (2001), no. 2, [CL04], Singularity of mean curvature flow of Lagrangian submanifolds, Invent. Math. 156 (2004), no. 1, [CLT02] J. Chen, J. Li, and G. Tian, Twodimensional graphs moving by mean cur 27
Extrinsic 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 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 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 informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
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 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 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 informationMotion by mean curvature and levelset approach
Motion by mean curvature and levelset approach Olivier Ley Laboratoire de Mathématiques et Physique Théorique Université de Tours Parc de Grandmont, 37200 Tours, France http://www.phys.univtours.fr/~ley
More informationDEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES. and MARCO ZAMBON
Vol. 65 (2010) REPORTS ON MATHEMATICAL PHYSICS No. 2 DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES IVÁN CALVO Laboratorio Nacional de Fusión, Asociación EURATOMCIEMAT, E28040 Madrid, Spain
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova GeometryTopology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
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 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 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 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 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 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 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 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 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 informationEikonal Slant Helices and Eikonal Darboux Helices In 3Dimensional Riemannian Manifolds
Eikonal Slant Helices and Eikonal Darboux Helices In Dimensional Riemannian Manifolds Mehmet Önder a, Evren Zıplar b, Onur Kaya a a Celal Bayar University, Faculty of Arts and Sciences, Department of
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 informationNonzero degree tangential maps between dual symmetric spaces
ISSN 14722739 (online) 14722747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris
More informationFiber Bundles and Connections. Norbert Poncin
Fiber Bundles and Connections Norbert Poncin 2012 1 N. Poncin, Fiber bundles and connections 2 Contents 1 Introduction 4 2 Fiber bundles 5 2.1 Definition and first remarks........................ 5 2.2
More informationBOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam
Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad
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 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 informationCARTAN S GENERALIZATION OF LIE S THIRD THEOREM
CARTAN S GENERALIZATION OF LIE S THIRD THEOREM ROBERT L. BRYANT MATHEMATICAL SCIENCES RESEARCH INSTITUTE JUNE 13, 2011 CRM, MONTREAL In many ways, this talk (and much of the work it reports on) owes its
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 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 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 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 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 informationWissenschaftliche Artikel (erschienen bzw. angenommen)
Schriftenverzeichnis I. Monographie [1] Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, BerlinHeidelberg New York, 2003.
More informationDimension Theory for Ordinary Differential Equations
Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskiinorms 15 1 Singular values
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 informationDavi Maximo Alexandrino Nogueira
Davi Maximo Alexandrino Nogueira 1 Education Department of Mathematics, Stanford University 450 Serra Mall, Bldg 380, Stanford, CA 94305 URL: http://math.stanford.edu/ maximo Email: maximo@math.stanford.edu
More informationThe cover SU(2) SO(3) and related topics
The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of
More informationON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE
Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 00192082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a
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 informationLECTURE 1: DIFFERENTIAL FORMS. 1. 1forms on R n
LECTURE 1: DIFFERENTIAL FORMS 1. 1forms on R n In calculus, you may have seen the differential or exterior derivative df of a function f(x, y, z) defined to be df = f f f dx + dy + x y z dz. The expression
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
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 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 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 informationSome ergodic theorems of linear systems of interacting diffusions
Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ liuyong@math.pku.edu.cn yangfx@math.pku.edu.cn 1 1 30 1 The ergodic theory of interacting systems has
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 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 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 informationSingular fibers of stable maps and signatures of 4 manifolds
359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a
More informationTensor product of vector spaces
Tensor product of vector spaces Construction Let V,W be vector spaces over K = R or C. Let F denote the vector space freely generated by the set V W and let N F denote the subspace spanned by the elements
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 informationLECTURE III. BiHamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland
LECTURE III BiHamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18 BiHamiltonian chains Let (M, Π) be a Poisson
More informationOn the degrees of freedom in shrinkage estimation
On the degrees of freedom in shrinkage estimation Kengo Kato Graduate School of Economics, University of Tokyo, 731 Hongo, Bunkyoku, Tokyo, 1130033, Japan kato ken@hkg.odn.ne.jp October, 2007 Abstract
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 informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationCLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS
POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 000 CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS PHILIPP E BO
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 informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
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 informationarxiv:math/0010057v1 [math.dg] 5 Oct 2000
arxiv:math/0010057v1 [math.dg] 5 Oct 2000 HEAT KERNEL ASYMPTOTICS FOR LAPLACE TYPE OPERATORS AND MATRIX KDV HIERARCHY IOSIF POLTEROVICH Preliminary version Abstract. We study the heat kernel asymptotics
More informationRICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS
RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS The Summer School consists of four courses. Each course is made up of four 1hour lectures. The titles and provisional outlines are provided
More informationRemarks on Lagrangian singularities, caustics, minimum distance lines
Remarks on Lagrangian singularities, caustics, minimum distance lines Department of Mathematics and Statistics Queen s University CRM, Barcelona, Spain June 2014 CRM CRM, Barcelona, SpainJune 2014 CRM
More informationThree observations regarding Schatten p classes
Three observations regarding Schatten p classes Gideon Schechtman Abstract The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation
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 informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationSix questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces
1 Six questions, a proposition and two pictures on Hofer distance for hamiltonian diffeomorphisms on surfaces Frédéric Le Roux Laboratoire de mathématiques CNRS UMR 8628 Université ParisSud, Bat. 425
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 informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationRESEARCH STATEMENT AMANDA KNECHT
RESEARCH STATEMENT AMANDA KNECHT 1. Introduction A variety X over a field K is the vanishing set of a finite number of polynomials whose coefficients are elements of K: X := {(x 1,..., x n ) K n : f i
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 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 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 informationLuminy Lecture 1: The inverse spectral problem
Luminy Lecture 1: The inverse spectral problem Steve Zelditch Northwestern University Luminy April 10, 2015 The inverse spectral problem The goal of the lectures is to introduce the ISP = the inverse spectral
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
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 informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
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 informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
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 informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationSmallest area surface evolving with unit areal speed
Smallest area surface evolving with unit areal speed Constantin Udrişte, Ionel Ţevy Abstract. The theory of smallest area surfaces evolving with unit areal speed is a particular case of the theory of surfaces
More informationSpectral Networks and Harmonic Maps to Buildings
Spectral Networks and Harmonic Maps to Buildings 3 rd Itzykson Colloquium Fondation Mathématique Jacques Hadamard IHES, Thursday 7 November 2013 C. Simpson, joint work with Ludmil Katzarkov, Alexander
More informationDIVISORS AND LINE BUNDLES
DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function
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 informationAlmost Quaternionic Structures on Quaternionic Kaehler Manifolds. F. Özdemir
Almost Quaternionic Structures on Quaternionic Kaehler Manifolds F. Özdemir Department of Mathematics, Faculty of Arts and Sciences Istanbul Technical University, 34469 MaslakIstanbul, TURKEY fozdemir@itu.edu.tr
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 informationIndex notation in 3D. 1 Why index notation?
Index notation in 3D 1 Why index notation? Vectors are objects that have properties that are independent of the coordinate system that they are written in. Vector notation is advantageous since it is elegant
More informationWhy do mathematicians make things so complicated?
Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010 Introduction What is Mathematics? Introduction What is Mathematics? 24,100,000 answers from Google. Introduction
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 informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationTensors on a vector space
APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,
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 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 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 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 informationTHE MINIMAL GENUS PROBLEM
THE MINIMAL GENUS PROBLEM TERRY LAWSON Abstract. This paper gives a survey of recent work on the problem of finding the minimal genus of an embedded surface which represents a twodimensional homology
More information