The Structure of Operators on Manifolds with Polyhedral Singularities

Propagation of singularities for the wave equation on edge manifolds

In this paper, we investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities.

Lie流形上的Sobolev空间与多面体域的正则性

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

本文研究Lie流形上的Sobolev空间与椭圆正则性，并应用于多面体域，证明强椭圆系统解无加权正则性损失，同时获得非标准边值问题的适定性。

Bounded $H_{\infty}$-calculus for Boundary Value Problems on Manifolds with Conical Singularities

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.

Propagation of singularities for the wave equation on manifolds with corners

In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-Δ_M, and u locally in H^1 solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions, we show that the wave front set of u is a union of maximally extended generalized broken bicharacteristics. This result is a smooth counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary. Our methods re…

Existence and maximal $L^{p}$-regularity of solutions for the porous medium equation on manifolds with conical singularities

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness and maximal $L^{p}$-regularity of a short time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin-Sobolev spaces and $R$-sectoriality perturbation techniques.

来自3维流形的具有预定奇点的热流

Heat Flows with Prescribed Singularities from 3-dimensional Manifold

本文研究了从三维完备有界黎曼流形到双曲空间且沿闭曲线具有预定奇点的奇异热流，证明了其存在性、正则性，并得出其以指数速率收敛到奇异调和映射的结论。

带锥奇点的流形上Cahn-Hilliard方程的全局吸引子存在性与解的收敛性

Existence of global attractors and convergence of solutions for the Cahn-Hilliard equation on manifolds with conical singularities

本文研究了带锥奇点的流形上的Cahn-Hilliard方程，证明了在带渐近性的高阶Mellin-Sobolev空间中全局吸引子的存在性及解向平衡点的收敛性，并给出了平衡点在锥尖附近的渐近行为。

The iterative Structure of Corner Operators

We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference "Elliptic and Hyperbolic Equations on Singular Spaces", October 27 - 31, 2008, at the MSRI, University of Berkeley.

Removability of singularities of harmonic maps into pseudo-Riemannian manifolds

We consider harmonic maps into pseudo-Riemannian manifolds. We show the removability of isolated singularities for continuous maps, i.e. that any continuous map from an open subset of R^m into a pseudo-Riemannian manifold which is two times continuously differentiable and harmonic everywhere outside an isolated point is actually smooth harmonic everywhere.

Some remarks on contact manifolds, Monge-Ampere equations and solution singularities

We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on some aspects of contact geometry.

Propagation of singularities under Schrödinger equations on manifolds with ends

We prove a microlocal smoothing effect of Schrödinger equations on manifolds. We employ radially homogeneous wavefront sets introduced by Ito and Nakamura (Amer. J. Math., 2009). In terms of radially homogeneous wavefront sets, we can apply our theory to both of asymptotically conical and hyperbolic manifolds. We relate wavefront sets in initial states to radially homogeneous wavefront sets in states after a time development. We also prove a relation between radially homogeneous wavefront sets and homogeneous wavefront sets and prove a special case of Nakamura (2005).

Perelman's Entropies for Manifolds with conical Singularities

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in space and time, where the maximal $L^{q}$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities

We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the Lp-setting. As a consequence we also obtain the spectral invariance of the classical Boutet de Monvel algebra of zero order operators with parameters. In order to establish these results, we show the equivalence of Fredholm property and ellipticity for both cases.

Complex powers for cone differential operators and the heat equation on manifolds with conical singularities

We obtain left and right continuous embeddings for the domains of the complex powers of sectorial $\mathbb{B}$-elliptic cone differential operators. We apply this result to the heat equation on manifolds with conical singularities and provide asymptotic expansions of the unique solution close to the conical points. We further show that the decomposition of the solution in terms of asymptotics spaces, i.e. finite dimensional spaces that describe the domains of the integer powers of the Laplacian and determined by the local geometry around the singularity, is preserved under the evolution.

The diffractive wave trace on manifolds with conic singularities

Let $(X,g)$ be a compact manifold with conic singularities. Taking $Δ_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{ \smash[b]{Δ_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces wit…

Generalized Dirac operators on Lorentzian manifolds and propagation of singularities

We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.

Elliptic $G$-operators on manifolds with isolated singularities

In the present work we study elliptic operators on manifolds with singularities in the situation, when the manifold is endowed with an action of a discrete group $G$. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair, consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). Fredholm property of elliptic elements is obtained.

具有锥奇点的流形上 Cahn-Hilliard 方程解的光滑性与长时间存在性

Smoothness and long time existence for solutions of the Cahn-Hilliard equation on manifolds with conical singularities

本文研究了锥奇点流形上的 Cahn-Hilliard 方程，证明了解在极大 $L^q$-正则性空间中的长时间存在性与瞬时光滑性，并刻画了解在锥尖附近的渐近行为。

Elliptic operators on manifolds with singularities and K-homology

It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula …

Heat trace expansion on manifolds with conic singularities

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry of the manifold.

A Cheeger-Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-Müller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

Diffraction of singularities for the wave equation on manifolds with corners

We consider the fundamental solution to the wave equation on a manifold with corners of arbitrary codimension. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners (i.e., loosely speaking, are not propagated along limits of transversely reflected rays) are smoother than the main singularities of the solution. More generally, we show that subject to a hypothesis of nonfocusing, diffracted wavefronts of any solution to the wave equation are smoother than the incident singularities. These results extend our previous work…

Propagation of singularities for the wave equation on conic manifolds

For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.

Singularities of mean convex level set flow in general ambient manifolds

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White and Haslhofer-Kleiner, and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy problem for the inhomogeneous heat equation, when the free term lies in a weighted parabolic Hölder or Sobolev space with discrete asymptotics. This generalizes a result previously obtained by Coriasco, Schrohe, and Seiler (Thm. 7.2 in Math. Z. 244 (2003), 235--269) by different mean…

在带锥形奇点的偶数维流形上指定$Q$-曲率

Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities

该研究针对带锥形奇点的$2m$维闭流形的指定$Q$-曲率问题，通过爆破分析与极小极大变分论证，得到超临界情形下的存在性与多重性结果，是$m>1$时非球面超临界锥形流形的首个相关存在性结论。

Families index for manifolds with hyperbolic cusp singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered b…

Desingularization of G_2 manifolds with isolated conical singularities

We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the resulting compact smooth 7-manifold admits a torsion-free G_2 structure, with full G_2 holonomy. There are topological obstructions for this procedure to work, which arise from the degree 3 and degree 4 cohomology of the asymptotically conical G_2 manifolds which are glued in at each c…

Singularities of solutions to Schrodinger equation on scattering manifold

In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition. Our result is closely related to a recent work by Hassell and Wunsch, though our model is more general and the method, which relies heavily on scattering theoretical ideas, is simple and quite different. In particular, we use Egorov-type argument in the standard pse…

光滑流形中平均凸平均曲率流的后续奇点

Subsequent singularities of mean convex mean curvature flows in smooth manifolds

本文证明在光滑流形中，若平均曲率流收敛到光滑超曲面，则其所有奇点均为圆柱型，推广了此前对维数或首奇点时刻的限制。

Yamabe metrics on cylindrical manifolds

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We affirmatively solve this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almos…