Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. TComment: 25 pages, 6 figue

Wedge product theorem in compensated compactness theory with critical exponents on Riemannian manifolds

We formulate and prove compensated compactness theorems concerning the limiting behaviour of wedge products of weakly convergent differential forms on closed Riemannian manifolds. The case of critical regularity exponents is considered, which goes beyond the regularity regime entailed by H\"{o}lder's inequality. Implications on the weak continuity of $L^p$-extrinsic geometry of isometric immersions of Riemannian manifolds are discussed.Comment: 25 page

ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY

This thesis aims to discuss some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. In this respect, we prove new results in both directions. For instance, we improve on classical oscillation and nonoscillation criteria for ODE's, and we find sharp spectral estimates for a number of geometric differential operator on Riemannian manifolds. We apply these results to achieve topological and geometric properties. In the first part of the thesis, we collect some material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view

Stochastic flows on noncompact manifolds

Here we look at the existence of solution flows of stochastic differential equations on noncompact manifolds and the properties of the solutions in terms of the geometry and topology of the underlying manifold itself. We obtain some results on "strong p-completeness” given conditions on the derivative flow, and thus given suitable conditions on the coefficients of the stochastic differential equations. In particular a smooth flow of Brownian motion exists on submanifolds of Rn whose second fundamental forms are bounded. Another class of results we obtain is on homotopy vanishing given strong moment stability. We also have results on obstructions to moment stability by cohomology. Also we obtain formulae for d(eᴢᵗΔʰØ) for differential form Ø in terms of a martingale and the form itself, not just its derivative, extending Bismut’s formula

Advances in Discrete Differential Geometry

Differential Geometr

Advances in Discrete Differential Geometry

Differential Geometr