DOCSLIB.ORG

Handle Crushing Harmonic Maps Between Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Handle Crushing Harmonic Maps Between Surfaces RICE UNIVERSITY Handle crushing harmonic maps between surfaces by Andy C. Huang A Tupsrs Sue\¿Irrpt rN Penuel FUIpILLMENT oF THE RnqurneMENTS FoR THE Dpcnpp Doctor of Philosophy AppRovpo, Tunsrs Covvtlrrpp: Michael Wolf, Professor of Mathematics Robert M. Hardt, W.L. Moody Professor of Mathematics Béatrice M. R,ivière, Professor of Computational and Applied Mathematics, Noah G. Harding Chair HousroN, TnxRS FpsRuRRv, 2076 Abstract Handle crushing harmonic maps between surfaces by Andy C. Huang In this thesis, we construct polynomial growth harmonic maps from once-punctured Rie- mann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyper- bolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from C (which are conformally once-punctured spheres) to H2 have been parameterized by holomorphic quadratic differentials on C ([WA94], [HTTW95], [ST02]). Our harmonic maps, mapping a genus g > 1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k + 2) at the puncture and (4g + k − 2) zeros (counting multiplicity). In this way, we can associate to these maps a holomorphic quadratic differential on the punctured Rie- mann surface domain. As an example, we explore a special case of our theorems: the unique harmonic map from a punctured square torus to an ideal square. We use the symmetries of the map to deduce the three possibilities for its Hopf differential. i Acknowledgments I owe so much to my parents, whose unconditional love and support without which I would not have gotten to where I am today. For everything they have provided me, and especially for encouraging me to be ever curious, I am thankful. I’d like thank my thesis advisor, Dr. Mike Wolf, for persuading me to “think like a harmonic map” when I needed inspiration, enduring many long discussions, and parsing my scribbles and drawings on his chalkboard. Thanks also to my thesis defense committee members Dr. Robert Hardt, for his kindness and mentorship, and Dr. Beatrice Riviere, who showed me an appreciation of numerical methods for thinking about PDEs, and the math department staff Marie Magee, Ligia Leismer, and Bonnie Hausman for pleasant conversations, administrative help, and encouragement. My academic siblings Qiongling Li and Jorge Acosta have been tirelessly enthusiastic and endlessly helpful. Another academic sibling, Zeno Huang, felt at times more like a second advisor and faithful supporter even though he was so far away. Without these people, my academic experience at Rice would have been incomplete. The graduate program has allowed me to experience so much and connect personally with so many. In no particular order, I thank Diego Vela, Timur Takhtaganov, Charles Puelz, and too many ultimate frisbee, climbing, cycling, and Rice Bikes shop friends to name here, for sharing their time with me in grad school, without whom I’d probably finish sooner but in poorer mental and physical health. ii Contents Abstract . .i Acknowledgments . ii List of Figures . vii 1 Introduction 1 1.1 Introduction and Main results . .1 1.1.1 Background . .5 1.2 Overview of thesis . 12 2 Mathematical preliminaries 13 2.1 Riemannian geometry concepts . 13 2.1.1 Tensors . 14 2.1.2 Connections . 17 2.1.3 Metrics on vector bundles with connections . 19 2.2 Harmonic map formulae . 20 2.2.1 The Dirichlet energy . 20 2.2.2 Existence and uniqueness . 23 2.2.3 Examples of harmonic maps . 25 2.2.4 Properties of harmonic maps . 26 2.3 Harmonic maps from surfaces . 27 iii 2.3.1 Regularity . 28 2.3.2 Specialization of harmonic map formulae . 29 2.3.3 Conformal invariance of energy . 30 2.3.4 Hopf differential . 31 2.3.5 Courant-Lebesgue Lemma . 34 2.3.6 Bochner formulae and consequences . 36 2.4 Some recent results on harmonic maps . 37 2.4.1 Harmonic maps C ! H2 ......................... 37 2.4.2 Wood’s analysis of images of harmonic maps . 38 2.4.3 Dumas-Wolf’s uniqueness of orientation-preserving harmonic maps through Hopf differentials . 39 2.4.4 Cheng’s Lemma . 40 2.4.5 Choe’s iso-energy inequality . 41 3 Handle-crushing harmonic maps 42 3.0.6 Organization of section . 42 3.1 Construction: Proof of Theorem 1.1.1 . 45 3.1.1 Set-up for construction . 46 3.1.2 Conditions for sub-convergence . 50 3.1.3 A comparison map for hs away from the handles . 55 3.1.4 Putting it all together . 59 3.2 Uniqueness: Proof of Theorem 1.1.2 . 61 3.3 Example: punctured square torus squashed onto ideal square . 63 3.4 The toy problem and The Energy Estimate . 66 3.4.1 Parachute maps . 69 3.4.2 Geometry of images . 70 iv 3.4.3 Hopf differentials of parachute maps . 71 3.4.4 Bounded Core Lemma . 83 3.4.5 The Energy Estimate: Parachute maps tied to Scherk map . 86 4 Future directions 90 4.1 The whole picture: handle crushers between compact surfaces . 91 4.1.1 Limit of compact maps to a diffeomorphism and a handle crushing model . 91 4.1.2 Bridging harmonic diffeomorphisms with handle crushers . 92 4.2 Minimal surfaces in H2 × R ............................ 94 4.3 Deforming handle crushers . 95 4.3.1 Deformations changing the image polygon . 95 4.3.2 Deformations preserving the image set . 95 4.4 Combinatorial data from Hopf differential . 96 4.5 Germs of harmonic maps . 98 4.5.1 Space of harmonic polynomials . 101 5 Tangential questions 103 5.1 Surface group representation domination by Fuchsian representations . 103 5.2 Discrete, non-faithful representations and branched hyperbolic structures . 104 A Riemannian geometry computations 105 A.1 Concerning Hopf differentials . 105 A.2 Geometry of harmonic mappings via their Hopf differentials . 107 B Holomorphic quadratic differential structures 109 B.1 Measured foliations . 109 B.2 Harmonic maps to R-trees . 111 v C Minimal suspensions in H2 × R 113 vi List of Figures 1.1 The possible Hopf differential configurations for Corollary 1.1.3 . .4 1.2 Approximate minimal suspension of map from Corollary 1.1.5.∗ .......6 h 1.3 Arrangement (a) horizontal and vertical foliations for F on S1 .......6 3.1 Identification space model for Sg ......................... 47 3.2 The Scherk map w : C ! H2 with Fw = −z2dz2................. 48 3.3 Example Pk for k = 4, 6, and 8. 49 3.4 Lines of symmetries for h : S1 !P4 from Corollary 1.1.3. 64 h 3.5 Possible locations for zeroes of F of h : S1 !P4 handle crusher. 64 4.1 Harmonic self map of unit disk exhibiting a cusp. 100 B.1 Foliations for Fu = z2dz2 on C ........................... 110 C.1 Suspension of u : C ! H2 with Fu(z) = z2dz2.y ................ 115 ∗Produced using Ken Brakke’s Surface Evolver program. yProduced using Ken Brakke’s Surface Evolver program. vii Chapter 1 Introduction 1.1 Introduction and Main results This thesis begins an investigation of the question: what is the shape of a harmonic map between surfaces from higher topological complexity to lower topological complexity? In order to focus this question, we consider the special class of harmonic maps between com- pact hyperbolic surfaces of different genera. This family of harmonic maps has the ap- pealing property that they are in bijection with homomorphisms between fundamental groups. ¥ For, any smooth map u 2 C (Sg, Sh) between compact hyperbolic surfaces (Sg, s) and (Sh, r) of any genera induces a homomorphism u∗ : p1(Sg) ! p1(Sh) between their fundamental groups. Conversely, since compact surfaces are K(p1, 1) spaces, any homo- morphism m between their fundamental groups can be induced by a continuous map f between them, i.e., f∗ ≡ m. Negative curvature of the target metric r ensures a unique harmonic representative exists in the homotopy class of that continuous map, if the map is non-constant. In the case the map is constant, the trivial homomorphism u∗ p1(Sg) = e is induced. We state this correspondence as ¥ 1−1 f[u] ju 2 C (Sg, Sh), harmonic, non-constantg ! u∗ 2 Hom p1(Sg), p1(Sh) n feg . 1 ∗ Furthermore, any smooth map u defines a symmetric 2-tensor u (r) on Sg by pulling back the target metric. We can use the complex structure induced by s on Sg to extract the (2, 0)-component of this pullback metric, called the Hopf differential of u, ( ) Fu := (u∗(r)) 2,0 . Hopf observed that the harmonicity of u implies holomorphicity of Fu with respect to this complex structure [Hop51]. Thus, there is a map ¥ u u 2 C (Sg, Sh) −! F 2 QD(Sg), where QD(Sg) is the vector space of holomorphic quadratic differentials on Sg. In this way, the complex analytic object Fu captures some of the differential topology of the harmonic map u and equivalently the algebraic information of the homomorphism u∗ between their fundamental groups p1(Sg) and p1(Sh). It is our present aim to create a local model to study the handle-crushing harmonic maps from higher genus to lower genus surfaces, and then characterize their Hopf dif- ferentials. Such handle crushing maps can be locally described as mapping a surface of positive genus and with boundary onto a disk, non-injectively taking interior to interior and monotonically taking boundary to boundary. These induce the trivial homomorphism on fundamental groups of punctured surfaces. We produce a model of handle crushing harmonic maps in the following: Theorem 1.1.1. (Existence) For any once punctured genus g ≥ 0 surface Sg and for any regular 2 2 ideal k-sided polygon Pk in H for k > 2 even, there exists a harmonic map h : Sg ! H whose image h(Sg) is the interior of Pk and whose closure h(Sg) = Pk.
Load more

Recommended publications
  • HARMONIC FUNCTIONS for BOUNDED and UNBOUNDED P(X)
    UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 Editor: Pekka Koskela Department of Mathematics and Statistics P.O. Box 35 (MaD) FI{40014 University of Jyv¨askyl¨a Finland ISBN 978-951-39-4299-1 ISSN 1457-8905 Copyright c 2011, Jukka Keisala and University of Jyv¨askyl¨a University Printing House Jyv¨askyl¨a2011 Contents 0 Introduction 3 0.1 Notation and prerequisities . 6 1 Constant p, 1 < p < 1 8 1.1 The direct method of calculus of variations . 10 1.2 Dirichlet energy integral . 10 2 Infinity harmonic functions, p ≡ 1 13 2.1 Existence of solutions . 14 2.2 Uniqueness of solutions . 18 2.3 Minimizing property and related topics . 22 3 Variable p(x), with 1 < inf p(x) < sup p(x) < +1 24 4 Variable p(x) with p(·) ≡ +1 in a subdomain 30 4.1 Approximate solutions uk ...................... 32 4.2 Passing to the limit . 39 5 One-dimensional case, where p is continuous and sup p(x) = +1 45 5.1 Discussion . 45 5.2 Preliminary results . 47 5.3 Measure of fp(x) = +1g is positive . 48 6 Appendix 51 References 55 0 Introduction In this licentiate thesis we study the Dirichlet boundary value problem ( −∆ u(x) = 0; if x 2 Ω; p(x) (0.1) u(x) = f(x); if x 2 @Ω: Here Ω ⊂ Rn is a bounded domain, p :Ω ! (1; 1] a measurable function, f : @Ω ! R the boundary data, and −∆p(x)u(x) is the p(x)-Laplace operator, which is written as p(x)−2 −∆p(x)u(x) = − div jru(x)j ru(x) for finite p(x).
    [Show full text]
  • Submanifolds of Euclidean Space with Parallel Second Fundamental Form
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 1, March 1972 SUBMANIFOLDS OF EUCLIDEAN SPACE WITH PARALLEL SECOND FUNDAMENTAL FORM JAAK VILMS1 In this paper necessary conditions are given for a complete Rie- mannian manifold M" to admit an isometric immersion into R*+r with parallel second fundamental form. Namely, it is shown that M" must be affinely equivalent either to a totally geodesic sub- manifold of the Grassmann manifold G(n,p), or to a fibre bundle over such a submanifold, with Euclidean space as fibre and the structure being close to a product. (An affine equivalence is a dif- feomorphism that preserves Riemannian connections.) The proof depends on the auxiliary result that the second fundamental form is parallel iff the Gauss map is a totally geodesic map. 1. The main result. Let i :Mn-+Rn+p be an immersion of a manifold Mn into Euclidean space, and endow M with the induced Riemannian metric. The second fundamental form ¡x of the immersion ; is a symmetric section of the bundle of bilinear maps L2(TM, TM; A/M), where TM and NM denote the tangent and normal bundles of M, respectively [4]. We iden- tify this bundle of maps with the isomorphic bundle L(TM, L(TM, NM)), and <xwith the corresponding section. At each point of M, the kernel of x (in the second interpretation) is called the relative nullity space, and its dimension k, the relative nullity index [2]. We say the second fundamental form a is parallel if 7>oc=0 (D always denotes the appropriate covariant derivative).
    [Show full text]
  • Berestycki, Introduction to the Gaussian Free Field and Liouville
    Introduction to the Gaussian Free Field and Liouville Quantum Gravity Nathanaël Berestycki [Draft lecture notes: July 19, 2016] 1 Contents 1 Definition and properties of the GFF6 1.1 Discrete case∗ ...................................6 1.2 Green function..................................8 1.3 GFF as a stochastic process........................... 10 1.4 GFF as a random distribution: Dirichlet energy∗ ................ 13 1.5 Markov property................................. 15 1.6 Conformal invariance............................... 16 1.7 Circle averages.................................. 16 1.8 Thick points.................................... 18 1.9 Exercises...................................... 19 2 Liouville measure 22 2.1 Preliminaries................................... 23 2.2 Convergence and uniform integrability in the L2 phase............ 24 2.3 The GFF viewed from a Liouville typical point................. 25 2.4 General case∗ ................................... 27 2.5 The phase transition in Gaussian multiplicative chaos............. 29 2.6 Conformal covariance............................... 29 2.7 Random surfaces................................. 32 2.8 Exercises...................................... 32 3 The KPZ relation 34 3.1 Scaling exponents; KPZ theorem........................ 34 3.2 Applications of KPZ to exponents∗ ....................... 36 3.3 Proof in the case of expected Minkowski dimension.............. 37 3.4 Sketch of proof of Hausdorff KPZ using circle averages............ 39 3.5 Proof of multifractal spectrum of Liouville
    [Show full text]
  • Book of Abstracts
    Georgian National Georgian Mathematical Ivane Javakhishvili Academy of Sciences Union Tbilisi State University Caucasian Mathematics Conference CMC I BOOK OF ABSTRACTS Tbilisi, September 5 – 6 2014 Organizers: European Mathematical Society Armenian Mathematical Union Azerbaijan Mathematical Union Georgian Mathematical Union Iranian Mathematical Society Moscow Mathematical Society Turkish Mathematical Society Sponsors: Georgian National Academy of Sciences Ivane Javakhishvili Tbilisi State University Steering Committee: Carles Casacuberta (ex officio; Chair of the EMS Committee for European Solidarity), Mohammed Ali Dehghan (President of Iranian Mathematical Society), Roland Duduchava (President of the Georgian Mathematical Union), Tigran Harutyunyan (President of the Armenian Mathematical Union), Misir Jamayil oglu Mardanov (Representative of the Azerbaijan Mathematical Union), Marta Sanz-Sole (ex officio; President of the European Mathematical Society), Armen Sergeev (Representative of the Moscow Mathematical Society and EMS), Betül Tanbay (President of the Turkish Mathematical Society). Local Organizing Committee: Tengiz Buchukuri (IT Responsible), Tinatin Davitashvili (Scientific Secretary), Roland Duduchava (Chairman), David Natroshvili (Vice chairman), Levan Sigua (Treasurer). Editors: Guram Gogishvili, Vazha Tarieladze, Maia Japoshvili Cover Design: David Sulakvelidze Contents Abstracts of Invited Talks 13 Maria J. Esteban, Symmetry and Symmetry Breaking for Optimizers of Functional Inequalities . 15 Mohammad Sal Moslehian, Recent Developments in Operator Inequalities . 15 Garib N. Murshudov, Some Application of Mathematics to Structural Biology 16 Dmitri Orlov, Categories in Geometry and Physics: Mirror Symmetry, D- Branes and Landau–Ginzburg Models . 17 Samson Shatashvili, Supersymmetric Gauge Theories and the Quantisation of Integrable Systems . 18 Leon A. Takhtajan, On Bott–Chern and Chern–Simons Characteristic Forms 19 Cem Yalçin Yildirim, Small Gaps Between Primes: the GPY Method and Recent Advancements Over it .
    [Show full text]
  • Properly-Weighted Graph Laplacian for Semi-Supervised Learning
    PROPERLY-WEIGHTED GRAPH LAPLACIAN FOR SEMI-SUPERVISED LEARNING JEFF CALDER Department of Mathematics, University of Minnesota DEJAN SLEPČEV Department of Mathematical Sciences, Carnegie Mellon University Abstract. The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit. In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement. 1. Introduction For many applications of machine learning, such as medical image classification and speech recognition, labeling data requires human input and is expensive [13], while unlabeled data is relatively cheap. Semi-supervised learning aims to exploit this dichotomy by utilizing the geometric or topological properties of the unlabeled data, in conjunction with the labeled data, to obtain better learning algorithms. A significant portion of the semi-supervised literature is on transductive learning, whereby a function is learned only at the unlabeled points, and not as a parameterized function on an ambient space. In the transductive setting, graph based algorithms, such the graph Laplacian-based learning pioneered by [53], are widely used and have achieved great success [3,26,27,44–48,50, 52].
    [Show full text]
  • Compact Foliations with Finite Transverse Ls Category 11
    COMPACT FOLIATIONS WITH FINITE TRANSVERSE LS CATEGORY STEVEN HURDER AND PAWELG. WALCZAK Abstract. We prove that if F is a foliation of a compact manifold M with all leaves compact submanifolds, and the transverse saturated category of F is finite, then the leaf space M=F is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. The transverse saturated category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with finite transverse saturated category. 1. Introduction A compact foliation is a foliation of a manifold M with all leaves compact submanifolds. For codimension one or two, a compact foliation F of a compact manifold M defines a fibration of M over its leaf space M=F which is a Hausdorff space, and has the structure of an orbifold [27, 11, 12, 33, 10]. A compact foliation F with Hausdorff leaf space is said to be compact Hausdorff. Millett [22] and Epstein [12] showed that for a compact Hausdorff foliation F of a manifold M, the holonomy group of each leaf is finite, a property which characterizes them among the compact foliations. If every leaf has trivial holonomy group, then a compact Hausdorff foliation is a fibration. Otherwise, a compact Hausdorff foliation is a \generalized Seifert fibration", where the leaf space M=F is a \V-manifold" [29, 17, 22]. In addition, M admits a Riemannian metric so that the foliation is Riemannian. For codimension three and above, the leaf space M=F of a compact foliation of a compact manifold need not be a Hausdorff space.
    [Show full text]
  • Descending Maps Between Slashed Tangent Bundles
    DESCENDING MAPS BETWEEN SLASHED TANGENT BUNDLES IOAN BUCATARU AND MATIAS F. DAHL f ABSTRACT. Suppose T M \ {0} and T M \ {0} are slashed tangent bundles of two smooth manifolds M and Mf, respectively. In this paper we characterize f those diffeomorphisms F : T M \ {0}→ T M \ {0} that can be written as F = f (Dφ)|TM\{0} for a diffeomorphism φ: M → M. When F = (Dφ)|TM\{0} one say that F descends. If M is equipped with two sprays, we use the charac- terization to derive sufficient conditions that imply that F descends to a totally geodesic map. Specializing to Riemann geometry we also obtain sufficient con- ditions for F to descent to an isometry. 1. INTRODUCTION In this paper we study the following differential-topological problem: ( ) Suppose M and M are smooth manifolds, and suppose that F is a diffeo- ∗ morphism betweenf slashed tangent bundles (1) F : T M 0 T M 0 . \ { } → f \ { } Characterize those maps F that can be written as F = (Dφ) for a |TM\{0} diffeomorphism φ: M M, where Dφ is the tangent map of φ. → f Problem ( ) is related to anisotropic boundary rigidity problems on Riemannian manifolds∗ [Cro04, Uhl01, PU05]. It is also the setting for studying conjugate geo- desic flows. For an overview of this topic for Riemann metrics, see [Ber07, p. 495]. When F = (Dφ) TM\{0} one say that map F descends to a map φ: M M [Cro04]. | → f Let us first note that if f is a diffeomorphism between (unslashed) cotangent bun- dles ∗ ∗ arXiv:0903.5133v3 [math.DG] 10 Jun 2009 f : T M T M, → f the analogous problem is well understood.
    [Show full text]
  • A Simple Discretization of the Vector Dirichlet Energy
    Eurographics Symposium on Geometry Processing 2020 Volume 39 (2020), Number 5 Q. Huang and A. Jacobson (Guest Editors) A Simple Discretization of the Vector Dirichlet Energy Oded Stein1, Max Wardetzky2, Alec Jacobson3 and Eitan Grinspun3;1 1Columbia University, USA 2University of Göttingen, Germany 3University of Toronto, Canada Abstract We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix-Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy. 1. Introduction The covariant derivative r generalizes the gradient of scalar func- tions to vector fields defined on surfaces. As the gradient does for scalar functions, the covariant derivative measures the infinitesimal change of a vector field in every direction. As with the gradient’s 1 R 2 scalar Dirichlet energy, Escalar(u) := 2 W kruk dx for a smooth scalar function u and a smooth surface W, the covariant derivative has a corresponding vector Dirichlet energy, Z 1 2 E(u) := krukF dx , (1) transporting the 2 W red vector (enlarged) denoising a vector field by smoothing across the surface where u is a smooth vector-valued function on W, and k·kF is the Frobenius norm.
    [Show full text]
  • The Laplacian in Its Different Guises
    U.U.D.M. Project Report 2019:8 The Laplacian in its different guises Oskar Tegby Examensarbete i matematik, 15 hp Handledare: Martin Strömqvist Examinator: Jörgen Östensson Februari 2019 Department of Mathematics Uppsala University Contents 1 p = 2: The Laplacian 3 1.1 In the Dirichlet problem . 3 1.2 In vector calculus . 6 1.3 As a minimizer . 8 1.4 In the complex plane . 13 1.5 As the mean value property . 16 1.6 Viscosity solutions . 17 1.7 In stochastic processes . 18 2 2 < p < and p = : The p-Laplacian and -Laplacian 20 2.1 Ded∞uction of th∞e p-Laplacian and -Laplaci∞an . 20 2.2 As a minimizer . ∞. 22 2.3 Uniqueness of its solutions . 24 2.4 Existence of its solutions . 25 2.4.1 Method 1: Weak solutions and Sobolev spaces . 25 2.4.2 Method 2: Viscosity solutions and Perron’s method . 27 2.5 In the complex plane . 28 2.5.1 The nonlinear Cauchy-Riemann equations . 28 2.5.2 Quasiconformal maps . 29 2.6 In the asymptotic expansion . 36 2.7 As a Tug of War game with or without drift . 41 2.8 Uniqueness of the solutions . 44 1 Introduction The Laplacian n ∂2u ∆u := u = ∇ · ∇ ∂x2 i=1 i � is a scalar operator which gives the divergence of the gradient of a scalar field. It is prevalent in the famous Dirichlet problem, whose importance cannot be overstated. It entails finding the solution u to the problem ∆u = 0 in Ω �u = g on ∂Ω. The Dirichlet problem is of fundamental importance in mathematics and its applications, and the efforts to solve the problem has led to many revolutionary ideas and important advances in mathematics.
    [Show full text]
  • Relations in Bounded Cohomology
    RELATIONS IN BOUNDED COHOMOLOGY JAMES FARRE Abstract. We explain some interesting relations in the degree three bounded coho- mology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degener- ate, while the other end is geometrically finite. We also show that a difference of two singly degenerate classes with bounded geometry is boundedly cohomologous to a dou- bly degenerate class, which has a nice geometric interpretation. Finally, we explain that the above relations completely describe the linear dependences between the `geometric' bounded classes defined by the volume cocycle with bounded geometry. We obtain a mapping class group invariant Banach sub-space of the reduced degree three bounded cohomology with explicit topological generating set and describe all linear relations. 1. Introduction The cohomology of a surface group with negative Euler characterisic is well understood. In contrast, the bounded cohomology in degree 1 vanishes, in degree 2, it is an infinite dimensional Banach space with the k · k1 norm [MM85, Iva88], and in degree 3 it is infinite dimensional but not even a Banach space [Som98]. In degree 4 and higher, almost nothing is known (see Remark 1.10). In this paper, we study a subspace in degree 3 generated by bounded fundamental classes of infinite volume hyperbolic 3-manifolds homotopy equivalent to a closed oriented surface S with negative Euler characteristic. These manifolds correspond to PSL2 C conjugacy classes of discrete and faithful representations ρ : π1(S) −! PSL2 C, but we restrict ourselves to the representations that do not contain parabolic elements to avoid technical headaches.
    [Show full text]
  • Arxiv:1703.06851V1
    SYMMETRIES AND CONSERVATION LAWS OF A NONLINEAR SIGMA MODEL WITH GRAVITINO JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU Abstract. We show that the action functional of the nonlinear sigma model with gravitino considered in [18] is invariant under rescaled conformal transformations, super Weyl transfor- mations and diffeomorphisms. We give a careful geometric explanation how a variation of the metric leads to the corresponding variation of the spinors. In particular cases and despite using only commutative variables, the functional possesses a degenerate super symmetry. The corre- sponding conservation laws lead to a geometric interpretation of the energy-momentum tensor and supercurrent as holomorphic sections of appropriate bundles. 1. Introduction The main motivation for the introduction of the two-dimensional nonlinear supersymmetric sigma model in quantum field theory, or more specifically super gravity and super string theory, are its symmetries, see for instance [3, 9]. Furthermore, as argued in [20], the functional is determined by its symmetries together with suitable bounds on the order of its Euler–Lagrange equations. While super symmetric models are usually formulated using anti-commutative vari- ables, in [18] an analogue of the two-dimensional nonlinear supersymmetric sigma model using only commutative variables was introduced. Here we would like to give a detailed geometric account of the symmetries of the purely commutative model considered in [18]. In physics as well as in geometry, symmetries of a system always have important implications. In particular, differentiable symmetries of a system give rise to conservation laws—this is ba- sically what the famous Noether’s theorem states, see e.g.
    [Show full text]
  • Regularity in the Calculus of Variations
    Regularity in the Calculus of Variations Connor Mooney 1 Contents 1 Introduction 3 2 Scalar Equations 4 2.1 Gradient Bounds Inherited From the Boundary . 7 2.2 De Giorgi-Nash-Moser . 8 2.3 Density Estimates and H¨olderRegularity . 10 2.4 Harnack Inequalities . 11 2.5 Boundary Estimates . 13 2.6 Dirichlet Problem for Minimal Surfaces . 16 2.7 Interior Gradient Estimate for Minimal Surfaces . 18 2.8 Optimal Gradient Bound . 22 2.9 Concluding Remarks on Minimal Surfaces . 24 2.10 Stability in the Calculus of Variations . 25 3 Elliptic Systems 27 3.1 Basics . 27 3.1.1 Existence . 27 3.1.2 Quasiconvexity . 28 3.1.3 Lower Semicontinuity . 29 3.1.4 Rank One Convexity . 31 3.2 Polyconvexity, Quasiconvexity, and Rank One Convexity . 32 3.2.1 Quadratic Functions . 32 3.3 Sverak's Example . 34 3.4 Linear Theory . 35 3.5 Partial Regularity . 37 3.6 Caccioppoli inequality for quasiconvex functionals . 39 3.7 Counterexamples . 40 3.8 Reverse H¨olderInequality . 42 3.9 Special Structure: Radial . 43 3.10 Quasiconformal Mappings . 44 2 1 Introduction In these notes we outline the regularity theory for minimizers in the calculus of variations. The first part of the notes deals with the scalar case, with emphasis on the minimal surface equation. The second deals with vector mappings, which have different regularity properties due to the loss of the maximum principle. These notes are based on parts of courses given by Prof. Ovidiu Savin in Fall 2011 and Spring 2014. 3 2 Scalar Equations Scalar divergence-form equations generally arise by taking minimizers of interesting energy functionals.
    [Show full text]