Motivated by the eigenvalue problems for ordinary and partial differential operators, we shall develop the spectral theory for linear operators in Hilbert spaces. Here we transform the unbounded differential operators into singular integral operators which are completely continuous. With his study of integral equations D. Hilbert, together with his students E. Schmidt, I. Schur, and H. Weyl, opened a new era for the analysis.

Linear Operators in Hilbert Spaces

This article considers the existence and regularity of Kahler{Einstein metrics on a compact Kahler manifold M with edge singularities with cone angle 2 along a smooth divisor D. We prove existence of such metrics with negative, zero and some positive cases for all cone angles 2 2 . The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coecients along D for all 2 < 2 .

Kahler-Einstein metrics with edge singularities

This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold $M$ with edge singularities with cone angle $2\pi\beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi\beta\leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi\beta < 2\pi$.

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.

Ricci flow on surfaces with conic singularities

We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrodinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case.

/pdf/on-higher-dimensional-singularities-for-the-fractional-2l73s710v1.pdf

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Let (M,g) be an odd-dimensional incomplete compact Riemannian singular space with a simple edge singularity. We study the analytic torsion on M, and in particular consider how it depends on the metric g. If g is an admissible edge metric, we prove that the torsion zeta function is holomorphic near s = 0, hence the torsion is well-defined, but possibly depends on g. In general dimensions, we prove that the analytic torsion depends only on the asymptotic structure of g near the singular stratum of M; when the dimension of the edge is odd, we prove that the analytic torsion is independent of the choice of admissible edge metric. The main tool is the construction, via the methodology of geometric microlocal analysis, of the heat kernel for the Friedrichs extension of the Hodge Laplacian in all degrees. In this way we obtain detailed asymptotics of this heat kernel and its trace.

Analytic Torsion on Manifolds with Edges

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.

Long-time existence of the edge Yamabe flow

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted product of zeta-regularized determinants of Laplace operators on differential forms. The celebrated Cheeger-Mueller Theorem, established independently by J. Cheeger and W. Mueller, proved equality between the analytic Ray-Singer torsion and the combinatorial Reidemeister torsion for any smooth closed manifold with an orthogonal representation of its fundamental group. Motivated by the vision of a Cheeger-Mueller type result on manifolds with conical singularities, we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of M. Spreafico and using the symmetry of the de Rham complex, as established by M. Lesch.

Analytic Torsion of a Bounded Generalized Cone

We discuss a specific class of regular-singular Laplace-type operators with matrix coefficients. Their zeta determinants were studied by K. Kirsten, P. Loya and J. Park on the basis of the Contour integral method, with general boundary conditions at the singularity and Dirichlet boundary conditions at the regular boundary. We complete the arguments of Kirsten, Loya and Park by explicitly verifying that the Contour integral method indeed applies in the regular-singular setup. Further we extend the zeta determinant computations to generalized Neumann boundary conditions at the regular boundary and apply our results to Laplacians on a bounded generalized cone with relative ideal boundary conditions.

Functional Determinants for Regular-Singular Laplace-type Operators

In this paper we study the analytic torsion of a manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of hig...

/pdf/the-metric-anomaly-of-analytic-torsion-on-manifolds-with-4gnehcqlr6.pdf

The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

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