Ph.D. Recipients and their Thesis Abstracts

Differential Geometry

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Existence and Compactness Theorems on Conformal Deformation of Metrics

Abstract: We prove that the set of solutions to the classical Yamabe equation, on a compact Riemannin n-manifold with positive Yamabe quotient, not necessarily locally conformally flat, is compact in the C^2 topology. Since we use the Positive Mass Theorem in the proof, we restrict ourselves to the cases 4 ≤ n ≤ 7. In the cases n = 6, 7, we also prove that the Weyl tensor has to vanish at a blowup point. The proofs are based on a careful blowup analysis of solutions. Given a compact n-manifold with umbilic boundary, n ≥ 9, finite Q (M, \partial M), such that the Weyl tensor does not vanish identically on \partial M, we show the existence of conformally related metrics with zero scalar curvature and constant mean curvature on \partial M. The proof of this result is based on an asymptotic analysis of the Sobolev quotients of explicitly defined test functions, using conformal Fermi coordinates.

On the Total Scalar Curvature Plus Total Mean Curvature Functional

Abstract: The total scalar curvature plus total mean curvature functional, defined on the space of Riemannian metrics of a compact manifold with boundary, arises in connection with the Yamabe problem for manifolds with boundary, its critical points on a given conformal class of metrics (subject to various volume and area constraints) being metrics of constant scalar curvature and constant mean curvature on the boundary. We characterize the critical points of this functional when the conformal class restriction is lifted, and the metrics are subject only to various volume and area constraints. We compute its second variation at critical points, and show that every critical point is a saddle point by giving examples of variations with positive second derivative (most conformal variations), zero second derivative (Lie derivatives of the metric) and negative second derivative (traceless tensor fields with positive dimensional null spaces). We show existence of minimizers on a given conformal class with Sobolev quotient strictly less that that of the upper half sphere, and show a compactness result for the set of all minimizers when metrics are allowed to vary on a small neighborhood of the space of metrics.

Abelian Sesquisymplectic Convexity for Orbifolds

Abstract: We show how the quotient of the symplectic normal bundle to a suborbifold of a symplectic orbifold can be viewed as the symplectic normal bundle to the image of the suborbifold in the reduction of the symplectic orbifold. We use this to prove an orbifold sesquisymplectic version of the isotropic embedding theorem. We apply this to prove a symplectic slice theorem in the same setting and in turn use this to prove a convexity theorem in this setting. Finally we apply these results to toric orbifolds.

On Conformal Metrics on the Euclidean Ball

Abstract: Not available.

Finiteness And Compactness For The Family Of Isospectral Riemannian Manifolds

Abstract: Riemannian manifolds are said to be isospectral if they have the same Laplacian spectrum by counting multiplicity. The purpose of this thesis is to study to what extent the Laplacian spectrum of a manifold M determines the manifold up to finitely many topological types, or up to a compact family of metrics. This research involves the studies of heat invariants, a-priori estimates for Ricci curvature equations, the properties of the space of isospectral Riemannian manifolds endowed with Gromov-Hausdorff or Lipschitz topology, and interpolation and embedding techniques, etc. Several finiteness and compactness results have been obtained with either geometric conditions or purely spectral conditions. One of the main theorems is as follows. Given constants p<= n and q> p/2. Let C^p_I(M) be the p-isoperimetric constant defined by

C^p_I(M)= inf_S (Area(S)) / ([min(Vol(A), Vol(B))]^{1–1/p}}),

for n<= p< \infty, where S runs over all hypersurfaces which divide M into two parts A and B. And let I(p,q) denote the space which consists of all closed, isospectral n-dimensional Riemannian manifolds with sectional curvatures uniformly bounded in L^q-norm and p-isoperimetric constants C^p_I(M) uniformly bounded away from zero. Then, I(p,q) contains only finitely many diffeomorphism types and it is compact in the C^\infty-topology, by which we mean that, on a fixed manifold in a given diffeomorphism class of I(p,q), the set of all pullback metrics of Riemannian metrics in the diffeomorphism class is compact with respect to the C^\infty-topology.