arXiv subjects

Mohammad Talebi

Publications and source records attributed to Mohammad Talebi.

4 recordsLinked to original sources

arXiv2025

Boundary Estimates for the Monge-Amp\`ere Equation in the Polygons with Guillemin Boundary Conditions

We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge--Amp\`ere equation on convex polytopes subject to the Guillemin boundary condition. Our result extends the work of Rubin and Huang to the case where the right-hand side is merely H\"older continuous. In particular, we obtain Euclidean $C^{1,\alpha/2}$ regularity up to the boundary, including along edges and at the vertices of the polytope, and then refine this to the sharp Euclidean $C^{1,\alpha}$ regularity. The analysis combines techniques introduced by Donaldson in his study of the Abreu equation with refined blow-up and localization arguments adapted to the degenerate boundary geometry.

math.AP↗

arXiv2021

Analytic torsion on manifolds with fibred boundary metrics

In this paper, we construct the renormalized analytic torsion in the setup of manifold endowed with fibred boundary metrics. The method of construction is to determine the asymptotic of heat kernel, both in short time regime and long time regime and apply these asymptotics together with renormalization to determine the renormalized zeta function and the determinant of Hodge Laplacian.

math.SP↗

arXiv2021

Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

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.

math.DG↗

arXiv2020

Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent

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.

math.DG↗