Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2{\pi} plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Wei{\ss} when the cone angles are less than {\pi}. We prove here an infinitesimal rigidity result valid for cone angles less than 2{\pi}, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author, see also the concurrent work by Wei{\ss}.Comment: 4 figures; minor modifications according to referee remarks. Accepted for publication in the Journal of Differential Geometr

Ricci flow on surfaces with conic singularities

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2\pi$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angle condition is satisfied (equivalently, when the data is logarithmically K-stable), the flow converges to the unique constant curvature metric with the given cone angles; if this condition is not satisfied, the flow converges subsequentially to a soliton. This is the one-dimensional version of the Hamilton--Tian conjecture.Comment: v1: 38 pages v2: 39 pages, restructured Sections 1 and 2, and added references and Subsection 5.4. v3-v4: 41 pages, revised to address referee comments; original proof of Proposition 5.3 had an error pointed out to us by a referee. We fix this by invoking Chow and Hamilton's original arguments instead of the Hamilton compactness theorem. Final version. To appear in Analysis and PD

Three circles theorems for harmonic functions

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.Comment: 34pp, to appear on Math. An

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

On the definition and examples of Finsler metrics

For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain of TM). Our aim is twofold. First, to give an account of quite a few subtleties which appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a previous step, the case of Minkowski conic pseudo-norms on an affine space). Second, to provide some criteria which determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler metric ---or, with more accuracy, the conic domain where g remains positive definite. Such a combination generalizes the known (alpha,beta)-metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors.Comment: 40 pages; v3: minor corrections, including Remark 2.9. To appear in Ann. Sc. Norm. Sup. Pis

Regular ambitoric 44-manifolds: from Riemannian Kerr to a complete classification

We show that the conformal structure for the Riemannian analogues of Kerr black-hole metrics can be given an ambitoric structure. We then discuss the properties of the moment maps. In particular, we observe that the moment map image is not locally convex near the singularity corresponding to the ring singularity in the interior of the black hole. We then proceed to classify regular ambitoric $4$-orbifolds with some completeness assumptions. The tools developed also allow us to prove a partial classification of compact Riemannian $4$-manifolds which admit a Killing $2$-form.Comment: 43 pages, 8 figures. Updated to fix a small error and extend the results to a broader class of metric

Long-time existence of the edge Yamabe flow

This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov-Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.Comment: 38 page

The harmonic map heat flow on conic manifolds

In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global solutions into manifolds with nonpositive sectional curvature. These results are established in virtue of the maximal regularity theory on manifolds with conic singularities

Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(\Delta_q+k^2)^{-1}$ of the Hodge Laplacian $\Delta_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+\delta$ and $\Delta_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+\delta)\Delta_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian manifolds $(\Omega_\epsilon,g_\epsilon)$ degenerating to a compact manifold $(\Omega_0,g_0)$ with a conic singularity as $\epsilon\to 0$.Comment: 53 pages, 3 figure

Conic geometric optimisation on the manifold of positive definite matrices

We develop \emph{geometric optimisation} on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (g-convex); and (ii) log-nonexpansive (LN). G-convex functions are nonconvex in the usual euclidean sense, but convex along the manifold and thus allow global optimisation. LN functions may fail to be even g-convex, but still remain globally optimisable due to their special structure. We develop theoretical tools to recognise and generate g-convex functions as well as cone theoretic fixed-point optimisation algorithms. We illustrate our techniques by applying them to maximum-likelihood parameter estimation for elliptically contoured distributions (a rich class that substantially generalises the multivariate normal distribution). We compare our fixed-point algorithms with sophisticated manifold optimisation methods and obtain notable speedups.Comment: 27 pages; updated version with simplified presentation; 7 figure