Related papers: Local Geometry of Singular Real Analytic Surfaces

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for…

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If…

We show how to define curvature as a measure using the Gauss-Bonnet Theorem on a family of singular surfaces obtained by gluing together smooth surfaces along boundary curves. We find an explicit formula for the curvature measure as a sum…

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $\sigma$. It generalizes Gromov's…

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated…

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary,…

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\RR^3$ with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in…

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…

Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holed torus (a sphere with $g$ handles). This paper is an introduction to the theory of compact Riemann surfaces and algebraic curves. It presents the basic ideas and…

We show that a locally symmetric space of noncompact type and with finite volume is quasi-isometric to the euclidean cone over a finite simplicial complex. A detailed analysis of metric properties yields a proof of a conjecture of Siegel.

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k)…

This article develops a unified and intrinsic framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is an explicit higher-order geometric integration by parts formula,…

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also,…

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for…

In this paper, we employ quotients of Roe algebras as index containers for elliptic differential operators to study the existence problem of Riemannian metrics with positive scalar curvature on non-compact complete Riemannian manifolds. The…

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,\mu}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $\Gamma$. We show that if $N$ is smooth, then $\Gamma$ is smooth and each $M_k$ is…

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…

We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a…

Hawking's local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full…