arXiv subjects

Aprameyan Parthasarathy

Publications and source records attributed to Aprameyan Parthasarathy.

7 recordsLinked to original sources

arXiv2025

Convexity on manifolds without focal points and applications

In this article, we study strictly convex functions on Riemannian manifolds without focal points, a broad class of manifolds encompassing all Hadamard manifolds as well as a large collection of manifolds whose sectional curvatures change sign. Using geometrically defined convex functions on such manifolds, we derive interesting consequences such as the continuity of the isoperimetric profile function without conditions on the sectional curvatures; if the manifolds are also K\"ahler, we obtain Steinness as well as a lower bound on the volume growth of metric balls. Our primary applications concern the spectrum of the Laplacian. We prove that the absolutely continuous part of the spectrum contains a certain infinite interval assuming only the existence of a point with respect to which the radial curvatures are nonpositive. This yields a generalization of the corresponding result for Hadamard manifolds. We use the geometry at infinity to give a new construction of a strictly convex function. We then apply this to show that the spectrum is purely absolutely continuous on a class of manifolds for which the horospheres in every direction at a single point have constant mean curvatures (e.g. asymptotically harmonic manifolds, symmetric spaces of noncompact type). Finally, we show the equality of Cheeger's constant and the volume entropy for a broad class of manifolds.

math.DG↗

arXiv2018

Boundary Values of Eigenfunctions on Riemannian Symmetric Spaces

We give a new and self-contained proof of a generic version of the (former) Helgason conjecture. It says that for generic spectral parameters the Poisson transform is a topological isomorphism, with the inverse given by a boundary value map. Following Oshima's approach to a simplified definition of boundary values, and using the earlier work of Baouendi and Goulaouic on Fuchsian type equations, our proof is along the lines of our earlier work in the rank one distributional case, and works for both the hyperfunction and the distribution setting.

math.RT↗

arXiv2017

Resonances and Scattering Poles in Symmetric Spaces of Rank One

We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of Kashiwara-Oshima. From this, we derive that the poles of the corresponding meromorphic continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp--Stein intertwiner. As a by-product of the underlying ideas, we obtain a new and self-contained proof of Helgason's conjecture for distributions in the case of rank one symmetric spaces.

math.SP↗

arXiv2016

Domains of holomorphy for irreducible admissible uniformly bounded representations of simple Lie groups

In this note, we address a question raised by Krötz on the classification of domains of holomorphy of irreducible admissible Banach representations for connected non-compact simple real Lie groups G. When G is not of Hermitian type, we give a complete description of the domains of holomorphy for irreducible admissible uniformly bounded representations on uniformly convex uniformly smooth Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine the domains of holomorphy only when the representations considered are highest or lowest weight representations.

math.RT↗

arXiv2015

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let $\mathbf{G}$ be a connected reductive complex algebraic group with split real form $(G,σ)$. Consider a strict wonderful $\mathbf{G}$-variety $\bf{X}$ equipped with its $σ$-equivariant real structure, and let $X$ be the corresponding real locus. Further, let $E$ be a real differentiable $G$-vector bundle over $X$. In this paper, we introduce a distribution character for the regular representation of $G$ on the space of smooth sections of $E$, and show that on a certain open subset of $G$ of transversal elements it is locally integrable and given by a sum over fixed points.

math.AG↗

arXiv2011

Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Regularized traces and characters

Consider a Riemannian symmetric space $X= G/K$ of non-compact type, where $G$ denotes a connected, real, semi-simple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $\widetilde X$ be its Oshima compactification, and $(π,\mathrm{C}(\widetilde X))$ the regular representation of $G$ on $\widetilde X$. In this paper, a regularized trace for the convolution operators $π(f)$ is defined, yielding a distribution on $G$ which can be interpreted as global character of $π$. In case that $f$ has compact support in a certain set of transversal elements, this distribution is a locally integrable function, and given by a fixed point formula analogous to the formula for the global character of an induced representation of $G$.

math.DG↗

arXiv2011

Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Microlocal analysis and kernel asymptotics

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 $G$, and characterize them within the framework of pseudodifferential operators, describing the singular nature of their kernels. In particular, we consider the holomorphic semigroup generated by a strongly elliptic operator associated to the representation $π$, as well as its resolvent, and describe the asymptotic behavior of the corresponding semigroup and resolvent kernels.

math.DG↗