arXiv research

Study resolvents of Bochner Laplacians on compact manifolds.

Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.

The study examines compact spaces resolvable by p-adic actions.

Krein's formula for conic Laplacians on compact Riemann surfaces

Estimates eigenvalues on differential forms on Alexandrov spaces.

Study on Dirac operator spectrum on shrinking surfaces with cusps.

The paper extends Witten's deformation to foliations and Morse functions.

In this paper we examine the Laplacian on the product of two asymptotically hyperbolic (or conformally compact, as they are often called) spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolv…

The study resolves a conjecture about harmonic forms on compact manifolds.

Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(π,\mathrm{C}(\widetilde \X))$ the regular representation of $G$ on $\widetilde \X$. We study integral operators on $\widetilde \X$ of the form $π(f)$, where $f$ is a rapidly falling function on $G$…

Proves stability of Einstein metrics on specific symmetric spaces.

We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through th…

For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.

We found unique tori with same curvatures using isometric transformations.

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. In a previous paper (Invent. math. 123 (1996), 507-552) the author constructed the first examples of compact 8-manifolds with holonomy Spin(7), by resolving orbifolds T^8/G, where T^8 is the 8-torus and G a finite group of automorphism…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=…

In this paper we consider certain asymptotically Euclidean spaces, namely compact manifolds with boundary X equipped with a scattering metric g, as defined by Melrose. We then consider Hamiltonians H which are `short-range' self-adjoint perturbations of the Laplacian of g. Melrose and Zworski have given a detailed desc…

Develops spectral triples for quantum projective space.

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of gr…

New local method solves Yamabe problems on compact and non-compact manifolds.

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ 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…

Sharp bounds on quasimode norms on compact space forms.

A new complex space resolves projective structures on surfaces.

This paper resolves equivariant K-theory for abelian actions.

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix i…

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing an…

Researchers resolve the compactness problem for helicoidal minimal surfaces in 3-sphere.

Resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial bundle smoothly decomposes into the direct sum of global one-dimensi…

The paper generalizes spectral section concepts to non-compact spaces.

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

Study the spectral properties of Laplacian on warped product manifolds.

New compact manifolds with closed G2 structures found.

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of …

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution…

Survey of analytic and geometric results on fibred cusp spaces.

The `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution tower' which projects …

Infinite diameter proved for contractible loops space.

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex …

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple descripti…

This paper constructs G$_2$ manifolds with nodal singularities along circles.

The study defines fields of definition for triangle groups as Fuchsian groups.

Study extends resolvent estimates for non-even metrics on hyperbolic spaces.

Develops Poisson and Dirac manifolds of compact types with applications.

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…

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution stru…

Formula for index of Dirac-type operators on stratified spaces.

Smooth submetries between curved spaces are smooth.

Study low energy resolvent behavior on fibred boundary metrics.