arXiv research

The paper proves properties of geometric flows on noncompact manifolds.

Study gives conditions for noncompact manifolds to have positive scalar curvature.

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odd-dimensional, compact Riemannian orbifold has a nont…

New uniformity definition on noncompact manifolds without metrics.

Optimizes heat equation estimates on noncompact manifolds.

Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.

Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.

Sharp inequalities on curved spaces with bounded curvature.

The study restricts scalar curvature on certain noncompact manifolds.

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to…

Paper proves a noncompact version of Gromov's band-width estimate.

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

In this paper, using the structures of cone and bicone fields on vector bundles, the author introduces a ILB (inverse limit of Banach)- manifold structure on $\mathcal M$ the space of Riemannian metrics on a noncompact manifold $M$. In the last section, it is proven that, this way, on the open submanifold $\mathcal{M}_…

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bund…

Extends Riemannian geometry inequalities with sharper estimates.

Sharp inequality found for hypersurfaces in curved spaces.

Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

Avoids noncompact hypersurfaces from touching in evolving flows.

An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As…

A foliation on a Riemannian manifold is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold that intersects each leaf of the foliation orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space of noncompact type.

Let $(M,g^{TM})$ be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and $F$ an integrable subbundle of $T M$ . Let $k^F$ be the leafwise scalar curvature associated to $g^F=g^{TM}|_F$. We show that if either $TM$ or $F$ is spin, then ${\rm inf}(k^F)\leq 0$. This gen…

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $\left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert Δu\right\Vert _{L^p},$ valid a priori for all smooth functions $…

The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

Paper finds periodic orbits for convex Lagrangian systems on noncompact manifolds.

We construct complete noncompact Riemannian metrics with $G_2$-holonomy on noncompact orbifolds that are $\Bbb R^3$-bundles with the twistor space $\mathcal{Z}$ as a spherical fiber.

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…

In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.

A theorem on odd dimensional noncompact manifolds shows curvature bounds.

Noncompact RCD spaces with maximal first Betti number are rigid.

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.

The study provides volume growth estimates for specific types of manifolds.

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic m…

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert $C^\ast$-module bundles and prove an index theorem for such operators…

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 …

The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

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.

In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.

Estimates submanifold diameters in curved spaces.

The paper is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in several aspects those obtained recently in the noncompact Riemannian setting.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n und…