arXiv research

Extends Llarull's theorem to noncompact manifolds with boundary.

We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…

New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.

The paper proves a new inequality for 3-manifolds with noncompact boundaries.

New theorem for spacetime mass in noncompact regions.

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …

Paper extends noncompact Llarull's theorem to manifolds with boundary.

Study mapping class groups of infinite type surfaces with noncompact boundaries.

The paper proves a theorem about splitting manifolds with specific curvature properties.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is …

Study finds conditions for minimal surfaces in noncompact spaces.

Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curv…

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…

The study classifies quasi-Einstein manifolds with constant scalar curvature.

New bounds for Dirac eigenvalue involving boundary capacity.

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several appl…

Estimates submanifold diameters in curved spaces.

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second fundamental form of the boundary. This extends to this setting previous results…

New invariant extends curvature estimates to noncompact manifolds.

Study on hyperbolic manifolds with special boundaries.

Study end sum for surfaces and prove uniqueness results.

Asymptotic symmetries of the five dimensional noncompact symmetric space SL(3)/SO(3) are found to form an infinite dimensional Lie algebra, analogously to the asymptotic symmetries of anti-de Sitter spaces in two and three dimensions. Possible exact solvability of the corresponding Chern-Simons theory and the AdS/CFT c…

The paper extends topological field theory to noncompact surfaces using symmetric powers.

Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a con…

Minimal surfaces span periodic curves in 3D space.

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of $p$. The special case of…

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

We give a short proof of the following fact. Let $Σ$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $Σ$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that i…

The paper studies curvature changes on manifolds with boundary.

The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary…

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitabl…

Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.

The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyp…

Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the hom…

Study of elliptic boundary value problems on non-compact manifolds.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

Paper proves new inequalities for Einstein-Maxwell data sets.

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $Φ$ is given by a power $β\geq 1$ of a monotone symmetric and homogeneous of degree one function $F$ of the principal curvatures. Under the assumption that $F$ …

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

Classifies surfaces for pure mapping class groups with automatic continuity.

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard …

In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.