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Using $\Gamma$-convergence, we study the Cahn-Hilliard problem with interface width parameter $\varepsilon > 0$ for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional exist and converge, as $\varepsilon \to 0$, to a function that takes only two values with an interface along a hypersurface that has minimal area among those satisfying a volume constraint. In a numerical example, we use continuation and bifurcation methods to study families of critical points at small $\varepsilon > 0$ on 2D elliptical cones, parameterized by height and ellipticity of the base. Some of these critical points are minimizers with interfaces crossing the cone tip. On the other hand, we prove that interfaces which are minimizers of the perimeter functional, corresponding to $\varepsilon = 0$, never pass through the cone tip for general cones with angle less than $2\pi$. Thus tip minimizers for finite $\varepsilon > 0$ must become saddles as $\varepsilon \to 0$, and we numerically identify the associated bifurcation, finding a delicate interplay of $\varepsilon > 0$ and the cone parameters in our example., Comment: 35 pages, 13 figures

We study the renormalized analytic torsion of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics. We establish invariance of the torsion under suitable deformations of the metric, and establish a gluing formula. As an application, we relate the analytic torsions for complete $\phi$- and incomplete wedge-metrics. As a simple consequence we recover a result by Sher and Guillarmou about analytic torsion under conic degeneration., Comment: 38 pages,10 figures

In this note we pursue a discrete analogue of a celebrated theorem by Osgood, Phillips and Sarnak, which states that in a fixed conformal class of Riemannian metrics of fixed volume on a closed Riemann surface, the zeta-determinant of the corresponding Laplace-Beltrami operator is maximized on a metric of constant Gaussian curvature. We study the corresponding question for the discrete cotan Laplace operator on triangulated surfaces. We show for some types of triangulations that the determinant (excluding zero eigenvalues) of the discrete cotan-Laplacian is stationary at discrete metrics of constant discrete Gaussian curvature. We present some numerical calculations, which suggest that these stationary points are in fact minima, strangely opposite to the smooth situation. We suggest an explanation of this discrepancy., Comment: 16 pages, 6 figures

Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_\Gamma$ be a Galois $\Gamma$-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^2$-Betti numbers and the Novikov-Shubin invariants are well defined. We then establish their invariance under a smoothly stratified, strongly stratum preserving homotopy equivalence, thus extending results of Dodziuk, Gromov and Shubin to these pseudo-manifolds., Comment: 33 pages, proof of Lemma 4.10 and Theorem 4.11 clarified, Corollary 5.3 improved

In this paper we equate the analytic and the intersection Reidemeister torsions on spaces with a specific type of wedge singularities, which arise by turning the disc cross-sections in the tubular neighborhood of an embedded submanifold of even co-dimension into cones. Our result is related to a similar equation, in a setting disjoint from ours, which was previously obtained by Albin, Rochon and Sher.

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics., Comment: 51 pages

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for m=2. Similar argument applies to higher dimensions as well. A conjecture on this asymptotics gives an equivalent formulation of the Epstein-Riemann conjecture, if we replace the standard discrete Laplacian with the $9$-point star discrete Laplacian., Comment: 37 pages, 2 figures

In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low energy condition. We also prove a concentration - compactness dichotomy, and investigate what the alternatives to convergence are. We end by investigating a non-convergent example due to Viaclovsky in more detail., Comment: 54 pages, 1 figure

The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In the case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Su\'arez-Serrato and Tapie to a non-compact setting. In the appendix, we specify our analysis of a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case, we obtain stronger estimates for the short-time solution using microlocal methods., Comment: 37 pages, 1 figure; v2: typos and reference corrected, v3: generalization to bounded geometry

In this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by Shi for complete manifolds, this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci de Turck flow for a short time., Comment: 42 pages, 2 figures

In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain conditions, Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow. We also discuss the relation to Perelman's entropies in the singular setting, and outline open questions and future reseach directions., Comment: 17 pages, 1 figure. arXiv admin note: text overlap with arXiv:1802.02908, arXiv:1902.02097, arXiv:2002.06586

In this paper we continue the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics, as initiated in our previous work. We consider the Hodge Laplacian for a $\phi$-metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners. Our discussion is a generalization of an earlier work by Albin and Sher, and provides a fundamental first step towards analysis of Ray-Singer torsion, eta-invariants and index theorems in the setting., Comment: 31 pages, 8 figures, minor revisions and corrections in the second version

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $\phi$-manifolds., Comment: minor corrections compared to the journal version, in particular in Theorem 8.4

In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature., Comment: 36 pages

Let M be a compact smoothly stratified pseudomanifold with boundary, satisfying the Witt assumption. In this paper we introduce the de Rham signature and the Hodge signature of M, and prove their equality. Next, building also on recent work of Albin and Gell-Redman, we extend the Atiyah-Patodi-Singer index theory established in our previous work under the hypothesis that M has stratification depth 1 to the general case, establishing in particular a signature formula on Witt spaces with boundary. In a parallel way we also pass to the case of a Galois covering M' of M with Galois group Gamma. Employing von Neumann algebras we introduce the de Rham Gamma-signature and the Hodge Gamma-signature and prove their equality, thus extending to Witt spaces a result proved by Lueck and Schick in the smooth case. Finally, extending work of Vaillant in the smooth case, we establish a formula for the Hodge Gamma-signature. As a consequence we deduce the fundamental result that equates the Cheeger-Gromov rho-invariant of the boundary of M' with the difference of the signatures of M and M'. We end the paper with two geometric applications of our results., Comment: 58 pages, 4 figures

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow., Comment: 34 pages, argument in Theorem 7.10 corrected

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$ is uniformly spacelike and asymptotic to $M\times\left\{s\right\}$ for some $s\in\mathbb{R}$ at infinity, we show that a mean curvature flow starting at $F_0$ exists for all times and converges uniformly to $M\times\left\{s\right\}$ as $t\to \infty$., Comment: 23 pages, final version

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., Comment: 41 pages

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of $\Delta$. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.

In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied., Comment: 48 pages, published version

We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition., Comment: revised; to appear in Journal of Spectral Theory

We consider Sturm-Liouville operators on a half line $[a,\infty), a>0$, with potentials that are growing at most quadratically at infinity. Such operators arise naturally in the analysis of hyperbolic manifolds, or more generally manifolds with cusps. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. Despite being the natural objects in the context of hyperbolic geometry, spectral geometry of such operators has only recently been studied in the context of analytic torsion., Comment: 31 pages

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 pages

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As an application, we discuss stability results for the two rho-invariants we have defined., Comment: 65 pages, 2 figures

In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The proof works by a careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation., Comment: 50 pages, 2 figures

We consider a combinatorial Laplace operator on a sequence of discrete graphs which approximates the m-dimensional torus when the discretization parameter tends to infinity. We establish a polyhomogeneous expansion of the resolvent trace for the family of discrete graphs, jointly in the resolvent and the discretization parameter. Based on a result about interchanging regularized limits and regularized integrals, we compare the regularized limit of the log-determinants of the combinatorial Laplacian on the sequence of discrete graphs with the logarithm of the zeta determinant for the Laplace Beltrami operator on the m-dimensional torus. In a similar manner we may apply our method to compare the product of the first N non-zero eigenvalues of the Laplacian on a torus (or any other smooth manifold with an explicitly known spectrum) with the zeta-regularized determinant of the Laplacian in the regularized limit as N goes to infinity., Comment: 18 pages

We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute explicitly the renormalized Ray-Singer analytic torsion of a model cusp in general dimension and without the Witt-condition. In the second step we establish a gluing formula for renormalized Ray-Singer analytic torsion on a general class of non-compact manifolds in any dimension that includes Witt-manifolds with cusps, but also scattering manifolds with asymptotically conical ends. In the final step, a Cheeger-Mueller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula., Comment: 64 pages, 2 figures

We define the combinatorial Dirichlet-to-Neumann operator and establish a gluing formula for determinants of discrete Laplacians using a combinatorial Gaussian quantum field theory. In case of a diagonal inner product on cochains we provide an explicit local expression for the discrete Dirichlet-to-Neumann operator. We relate the gluing formula to the corresponding Mayer-Vietoris formula by Burghelea, Friedlander and Kappeler for zeta-determinants of analytic Laplacians, using the approximation theory of Dodziuk. Our argument motivates existence of gluing formulas as a consequence of a gluing principle on the discrete level., Comment: 26 pages, accepted for publication at Letters in Math. Physics

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a unitary point. Finally we provide a new proof of Bruening-Ma gluing formula for the Ray-Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Bruening and Ma and uses a temporal gauge transformation., Comment: 37 pages, several important changes suggested by the referee

This is a continuation of the first author's development of the theory of elliptic differential operators with edge degeneracies. That first paper treated basic mapping theory, focusing on semi-Fredholm properties on weighted Sobolev and H\"older spaces and regularity in the form of asymptotic expansions of solutions. The present paper builds on this through the formulation of boundary conditions and the construction of parametrices for the associated boundary problems. As before, the emphasis is on the geometric microlocal structure of the Schwartz kernels of parametrices and generalized inverses., Comment: 45 pages, 3 figures

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revolution decomposes into an infinite direct sum of Sturm-Louville operators, parametrized by the eigenvalues of the Laplacian on the cross-section. We apply the polyhomogeneous expansion to equate the zeta-determinant of the Laplace-Beltrami operator as a regularized sum of zeta-determinants of the Sturm-Liouville operators plus a locally computable term from the polyhomogeneous resolvent trace asymptotics. This approach provides a completely new method for summing up zeta-functions of operators and computing the meromorphic extension of that infinite sum to s=0. We expect out method to extend to a much larger class of operators.

Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easily from the Cheeger-Mueller theorem, i.e. from the equality of the two torsions on the set of unitary representations. The proof uses the fact that the refined analytic torsion is a holomorphic function on the space of representations.

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. This yields short time existence for a class of semi-linear equations of fourth order, including for example the Cahn-Hilliard equation. We also obtain asymptotics of the solutions near the edge singularity., Comment: 23 pages, v2. extensive revision

We discuss the exotic properties of the heat-trace asymptotics for a regular-singular operator with general boundary conditions at the singular end, as observed by Falomir, Muschietti, Pisani and Seeley as well as by Kirsten, Loya and Park. We explain how their results alternatively follow from the general heat kernel construction by Mooers, a natural question that has not been addressed yet, as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non-polyhomogeneity of the heat kernel., Comment: 15 pages, 1 figure, v2: minor corrections

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setup of edges. We then establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in case of a non-Friedrichs extension., Comment: 31 pages, 3 figures, v3 extensive revision. Accepted for publication at Journal d'Analyse Mathematique

We establish multiparameter resolvent trace expansions for elliptic boundary value problems, polyhomogeneous both in the resolvent and the auxiliary parameter. The present analysis is rooted in the joint project with Matthias Lesch on multiparameter resolvent trace expansions on revolution surfaces with applications to regularized sums of zeta-determinants., Comment: Minor corrections after the referee report

Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain H\"older spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting., Comment: 3 pages, 1 figure, v2: minor improvements, references added, organizational changes

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya., Comment: 41 pages, 1 figure; v2: substantial additions, improvements, and organizational changes

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., Comment: 36 pages, 5 figures, v2: minor improvements

We consider Sturm-Liouville operators on the line segment [0, 1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski- determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials., Comment: 38 pages, 2 figures; Completely revised according to the referees comprehensive suggestions; v3: minor corrections, accepted for publication in Journal of Functional Analysis