Differential Geometry and Analysis Seminar

Schedule, Fall 2002

• Sept 18: Christina Sormani, Lehman College
  "The Stability of Isotropy and Schur's Lemma"
  
  Abstract: A Riemannian manifold is locally isotropic if locally the length of a side of a triangle depends only on the lengths of the other two sides, their common point, and the angle between them. By Schur's Lemma such a manifold is a space form (e.g. has constant sectional curvature). Here we will say that a Riemannian manifold is almost locally isotropic if locally the length of a side of a triangle is close to a function $F$ of the lengths of the other two sides, their base point and the angle between them. We show that if compact manifold is sufficiently almost isotropic depending on the minimum of its Ricci curvature then it is Gromov-Hausdorff close to a compact space form and the function $F$ is that of the space form.
  
  
• Sept 25: David Fisher, Lehman College
  "Local Rigidity of Isometric Actions."
  
  Click here for the abstract.
  
  
• Oct 2: Washek Pfeffer, University of California, Davis
  hosted by: Dennis Sullivan/Mehrzad Ajoodanian
  "Derivatives and Primatives"
  
  Abstract: A very general Gauss-Green theorem can be obtained by differentiating certain additive functions defined on the family of all bounded BV sets.  The process is invariant with respect to local lipeomorphisms, and the resulting integration by parts formula can be used to study removable sets for PDEs in divergence form.  I plan to explain these results in an elementary fashion, without assuming familiarity with BV sets and functions.
  
  
• Oct 9: Raphael Krikorian, Ecole Polytechnique, Paris
  hosted by: David Fisher
  "Positive Lyapunov exponents and differentiable rigidity of linear quasi-periodic cocycles: non perturbative results"
  
  Abstract: Let us consider linear quasi-periodic cocycles which define the dynamics on R/Z x SL(2,R) which act as shifts on the torus and smoothly on SL(2, R) by an action A. The results are then the following: If the action alpha on the torus is fixed and satisfies a diophantine like property (full measure condition) then the set of maps A for which the fibered exponent of (alpha,A(.)) is positive, is dense in the smooth topology.
  
  
• Oct 16: No meeting monday schedule
  
  
• Oct 23: Takayuki Yamauchi, DePauw University
  hosted by: John Velling
  "On the Construction of a Complete CMC Surface in the Hyperbolic 3-Space Determined by a Prescribed Hopf Differential"
  
  Abstract: The role that a prescribed holomorphic Hopf differential A(z)dzdz plays in the construction of a negatively curved immersed simply connected complete surface S0 of prescribed constant mean curvature c \in (-1, 1) in the hyperbolic 3-Space H3 is investigated in this work. When a holomorphic function A(z), which is the coefficient function of the Hopf differential, is prescribed on a unit disk |z| < 1, it is shown that the unit disk can be immersed in the hyperbolic 3-Space H3 as a negatively curved complete surface of constant mean curvature c \in (-1, 1), provided that |A(z)| satisfies a certain growth condition. Moreover, it is shown that the unit disk can be uniquely embedded in H3 when the holomorphic function A(z) has a certain admissible structure.
  
  
• Oct 30: Bruno Klingler, Yale
  hosted by: David Fisher
  "On the Arithmeticity of Some Complex Hyperbolic Lattices"
  Note the proof uses harmonic maps :) see the abstract.
  
  
• Nov 6: Xiaochun Rong, Rutgers University at New Brunswick
  hosted by: Christina Sormani
  "A connectedness principle in the geometry of positive curvature"
  
  We develop a connectedness principle in the geometry of positive curvature. In the form this is a surprising analog of the classical connectedness principle in complex algebraic geometry. The connectedness principle provides with not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking, but also new connectedness theorems of positive curvature.
  
  
• Nov 13: John Toth, McGill University
  "Riemannian manifolds with uniformly bounded eigenfunctions."
  Abstract: Let (M,g) be a compact, Riemannian manfold with Laplace- Beltrami operator, \Delta. Assume that the L^{2}-normalized Laplace eigenfunctions are all uniformly bounded. Does this imply that (M,g) is a flat torus? We discuss some recent joint work with Steve Zelditch that answers this question in the affirmative when \Delta is quantum completely integrable.
  hosted by: David Fisher and Yiannis Petrides
  
  
• Nov 20: Mu Tao Wang, Columbia University
  hosted by: David Fisher/Christina Sormani
  "The Dirichlet problem of the minimal surface system"
  
  
• Nov 27: Yiannis Petrides, Lehman College CUNY
  "On the spectrum of Heisenberg manifolds.
  Abstract: I will explain how methods from geometry (calculating the geodesics), analysis (Guillemin-Duistermaat trace-formula) and analytic number theory (Van der Corput's mathod on exponential sums and the method of exponent pairs) can provide interesting results on the distribution of eigenvalues of Heisenberg manifolds, in particular, on estimating the error term in Weyl's Law.
  
  
• Dec 4: Penelope Smith, Lehigh University
  "Perron's Method for Highly Nonlinear Second Order Hyperbolic Equations"