The ends of manifolds with bounded geometry, linear growth and finite filling area

Manifolds with quadratic curvature decay and fast volume growth

We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

关于具有线性体积增长和非负里奇曲率的流形在无穷远处的几何

On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature

本文证明了具有非负里奇曲率、线性体积增长和非塌缩端的开流形在无穷远处分裂出直线，从而完成大体积等周集存在性证明，并给出线性体积增长的等价刻画及截面非同胚的柱体极限空间例子。

The topology, geometry and conformal structure of properly embedded minimal surfaces

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff space, naturally ordered by the relative heights of the ends in space. One of our main results is that the middle ends of the surface have quadratic area growth, and are thus not limit ends. This implies, for instance, that the surface can have at most two limit ends (at the top and bottom of the ordering), which is a strong topological restriction. There are al…

Symplectic stability on manifolds with cylindrical ends

A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and compl…

On the flatness of Riemannian cylinders without conjugate points

What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by the length of horocycles. This is used to extend results by K. Burns and G. Knieper [9], and by H. Koehler [18], where the opening of the ends is measured in terms of shortest noncontractible loops.

Finite Morse index implies finite ends

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in $\R^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of cl…

关于某些有理数秩为一的算术群上同调中 torsion 的增长

On the growth of torsion in the cohomology of some arithmetic groups of $\mathbb{Q}$-rank one

本文研究了某些有理数秩为一的算术群的同调中 torsion 的增长，通过解析 torsion 与 Reidemeister torsion 的关系，证明了同余子群序列中 torsion 的指数增长。

The Almost Rigidity of Manifolds with Lower Bounds on Ricci Curvature and Minimal Volume Growth

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology. Manifolds with nonnegative Ricci curvature and lin…

Positive solutions to Schrödinger equations and geometric applications

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds …

Hodge theory and cohomology with compact supports

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_μ$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.

Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds…

Nilpotent groups without exactly polynomial Dehn function

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

Uniqueness of solutions, radiation conditions, and complexity of the metric at infinity

The purpose of this paper is to prove the uniqueness theorem of solutions of eigenvalue equations on one end of Riemannian manifolds for drift Laplacians, including the standard Laplacian as a special case; we shall impose "a sort of radiation condition" at infinity on solutions. We shall also provide several Riemannian manifolds whose Laplacians satisfy the absence of embedded eigenvalues and besides the absolutely continuity, although growth orders of their metrics on ends are very complicated.