Positivity Of Curvature Research Articles

Positivity of Curvature On Manifolds With Boundary

Abstract Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ that agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

A set $S\subseteq\mathbb{R}^n$ is called semidefinite programming (SDP) representable or semidefinite representable if S equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to constructing SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\dots,W_m$, we give an explicit construction of an SDP representation for $\mathrm{conv}(\cup_{k=1}^mW_k)$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary $\partial S$ is nonsingular and positively curved, while necessary is: $\partial S$ has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on $\partial S$ and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity at those points on $\partial S$ where they vanish. The gaps between them are $\partial S$ having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is $\partial_c T$, the maximum subset of $\partial T$ contained in $\partial\mathrm{conv}(T)$. We prove sufficient for SDP representability: $\partial_c T$ is nonsingular and positively curved, and necessary is: $\partial_c T$ has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.

Positivity of curvature and convexity of faces

Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.

A new Li-Yau-Hamilton estimate for the Ricci flow

Since its discovery, Hamilton’s Li–Yau–Hamilton (LYH) estimate has proven its importance in many different works (for example [7], [9], [10], [6]), and as a result, similar inequalities have subsequently appeared in the study of numerous other geometric flows – the mean curvature flow [11], the Kahler– Ricci flow [1], and the Gauss curvature flow [2], among others. The importance of LYH-type estimates is underscored by the fact that the discovery of an LYH estimate without curvature assumption is a large step in Hamilton’s program for Geometrization. Traditionally, positivity of curvature in some form has always been needed for the existence of an LYH estimate. However, work of Hamilton [10] and Ivey [13] in three dimensions indicate that because the curvature operator becomes (in a sense) close to positive near singularities, and because there is an LYH estimate for positive curvature operator, there should be an LYH estimate without any curvature assumptions. Because of this, one approach towards finding an inequality on spaces of arbitrary curvature is to perturb the LYH estimates that are found when there is positive curvature. In fact, using this point of view, an LYH estimate was discovered on surfaces where some negative curvature was allowed [12]. This provides great motivation to find and understand the LYH estimates that do exist when positivity is assumed, and to discover the deeper reasons why such inequalities exist. One very interesting approach towards understanding LYH estimates attempts to view these inequalities in a geometric setting. This was first accomplished in the work of Chow and Chu [3], where a degenerate metric and a space-time connection is place on the flow. In this setting, Hamilton’s original LYH quantity appears very naturally and geometrically, being closely related to the curvature of this conenction. In subsequent papers, Chow and Chu [4], as well as Chow and Knopf [5] have since gained more understanding of this point of view. They have been able to refine it, clarify-