《Handbook of Geometric Analysis》讲述了Geometric Analysis combines difierentiaI equations and difierential geometry．Animportant aspect iS to solve geometric problems by studying difierentiaI equations．Besides some known linear difierential operators such as the Laplace operator,many difierential equations arising from difrerential geometry are nonlinear．Aparticularly important example iS the Monge-Amp~~re equation。Applications togeometric problems have also motivated new methods and techniques in differen-tiaI equations。The field of geometric analysis iS broad and has had many strikingapplications．This handbook of geometric analysis provides introductions to andsurveys of important topics in geometric analysis and their applications to relatedfields which iS intend t0 be referred by graduate students and researchers in relatedareas． |
numerical approximations to extremal metrics on toric surfaces . r. s. bunch, simon k. donaldson 1 introduction 2 the set-up 3 numerical algorithms: balanced metrics and refined approximations 4 numerical results 5 conclusions references kahler geometry on toric manifolds, and some other manifolds with large symmetry simon k. donaldson introduction 1 background 2 toric manifolds 3 toric fano manifolds 4 variants of toric differential geometry 5 the mukai-umemura manifold and its deformations references gluing constructions of special lagrangian cones mark haskins, nikolaos kapouleas .i introduction 2 special lagrangian cones and special legendrian submanifolds of s(2n-1) 3 cohomogeneity one special legendrian submanifolds of s(2n-1) 4 construction of the initial almost special legendrian submanifolds 5 the symmetry group and the general framework for correcting the initial surfaces 6 the linearized equation 7 using the geometric principle to prescribe the extended substitute kernel 8 the main results a symmetries and quadratics references harmonic mappings jurcen jost 1 introduction 2 harmonic mappings from the perspective of riemannian geometry. 3 harmonic mappings from the perspective of abstract analysis and convexity theory 4 harmonic-mappings in kahler and algebraic geometry. 5 harmonic mappings and riemann surfaces references harmonic functions on complete riemannian manifolds peter li introduction 1 gradient estimates 2 green's function and parabolicity 3 heat kernel estimates and mean value inequality 4 harmonic functions and ends 5 stability of minimal 6 polynomial growth harmonic functions 7 massive sets and the structure of harmonic maps 8 lq harmonic functions references complexity of solutions of partial differential equations fang hua lin 1 introduction 2 level and critical point sets 3 solutions of nonlinear equations 4 a partition problem for eigenvalues acknowledgement references variational principles on triangulated surfaces feng luo 1 introduction 2 the schlaefii formula and its counterparts in dimension 2 3 variational principles on surfaces 4 the moduli spaces of polyhedral metrics 5 several open problems references asymptotic structures in the geometry of stability and extremal metrics toshiki mabuchi 1 extremal metrics in kahler geometry 2 stability for polarized algebraic manifolds 3 the asymptotic bergman kernel 4 test configurations 5 affine sphere equations 6 "affine spheres" for toric einstein surfaces 7 asymptotic expansion for toric einstein surfaces references stable constant mean curvature surfaces william h. meeks iii, joaqufn perez, antonio ros 1 introduction 2 stability of minimal and constant mean curvature surfaces .. 3 weak h-laminations 4 the stable limit leaf theorem 5 foliations by constant mean curvature surfaces 6 removable singularities and local pictures 7 compactness of finite total curvature surfaces 8 singular minimal laminations 9 the moduli space of embedded minimal surfaces of fixed genus 10 appendix references a general asymptotic decay lemma for elliptic problems leon simon introduction 1 scale invariant compact classes of submanifolds 2 some preliminaries concerning the class p 3 stability inequality 4 compact classes of cones 5 a partial harnack theory 6 proof of theorem 1 7 application to growth estimates for exterior solutions references uniformization of open nonnegatively curved kahler manifolds in higher dimensions luen-fai tam 1 introduction 2 function theory on kshler manifolds 3 busemann function and the structure of nonnegatively curved khhler manifolds 4 kahler-ricci flow 5 uniformization results references geometry of measures: harmonic analysis meets geometric measure theory tatiana. toro 1 introduction 2 density - an indicator of regularity 3 harmonic measure: boundary structure and size 4 geometric measure theory tools 5 open questions references the monge-ampere eequation and its geometric aapplications neil s. trudinger, xu-jia wang 1 introduction 2 the monge-ampere measure 3 a priori estimates 4 existence and uniqueness of solutions 5 the affine metric 6 affine maximal surfaces references lectures on mean curvature flows in higher codimensions mu-tao wang 1 basic materials 2 mean curvature flow 3 blow-up analysis 4 applications to deformations of symplectomorphisms of riemann surfaces 5 acknowledgement references local and global analysis of eigenfunctions on riemannian manifolds steve zelditch introduction 1 basic definitions and notations 2 explicitly solvable eigenfunctions 3 local behavior of eigenfunctions 4 nodal sets on c∞ riemannian manifolds 5 the wave kernel of a compact riemannian manifold 6 methods for global analysis 7 singularities pre-trace formulae 8 weyl law and local weyl law 9 local and global lp estimates of eigenfunctions 10 gaussian beams and quasi-modes associated to stable closed geodesics 11 birkhoff normal forms around closed geodesics 12 quantum integrable laplacians 13 concentration and non-concentration for general (m, g) 14 lp norms and concentration in the quantum integrable case 15 delocalization in quantum ergodic systems, i 16 delocalization of eigenfunctions: ii: entropy of quantum limits on manifolds with anosov geodesic flow 17 real analytic manifolds and their complexifieations 18 riemannian random waves 19 appendix on tauberian theorems references yau's form of schwarz lemma and arakelov inequality on moduli spaces of projective manifolds kang zuo introduction 1 polarized complex variation of hodge structure and higgs bundle 2 viehweg's positivity theorem for direct image sheaves 3 coverings, constructing higgs bundles and the positivity on 4 algebraic hyperbolicity and effective boundedness references |
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