arXiv subjects

Tirumala Chakradhar

Publications and source records attributed to Tirumala Chakradhar.

5 recordsLinked to original sources

arXiv2026

Lower bounds for the low Steklov eigenvalues

For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $σ_k$, $1\le k\le b-1$. Our results complement earlier results, which apply only to $σ_k$ with $k\ge b$ and depend on the geometry near the boundary, by showing how the interior geometry influences the low eigenvalues. Our result also yields lower bounds for the low Steklov eigenvalues in the setting of pinched negatively curved manifolds, thus recovering similar results in that context through an alternative proof. The proof of the main result is based on the trace inequality relating the Steklov eigenvalue to the Neumann eigenvalues of the connected subdomains of the manifold containing a boundary collar. The geometric coefficient appearing in this inequality is given by an explicit formula in terms of a quantity that can be interpreted as the electrical resistance of the boundary collar.

math.DG↗

arXiv2026

Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains

We establish geometric lower bounds for the smallest positive eigenvalue of the Hodge Laplacian in the class of non-convex domains given by Euclidean annular regions with a convex outer boundary and a spherical inner boundary. These bounds are then extended to convex domains with multiple holes, where we derive lower bounds for certain higher order exact eigenvalues, and under additional geometric assumptions, also for the smallest positive eigenvalue. For $1$-forms on compact manifolds with boundary, we provide a general lower bound on the smallest exact eigenvalue - corresponding to the first positive Neumann eigenvalue - which, in certain respects, is better than the classical Cheeger inequality. Furthermore, we emphasise the necessity of the "contact radius" in the lower bounds of the main results. Our proofs employ local-to-global arguments via an explicit isomorphism between Čech cohomology and de Rham cohomology to obtain Poincaré-type inequalities with explicit geometric dependence, and utilise certain generalised versions of the Cheeger-McGowan gluing lemma.

math.DG↗

arXiv2025

Magnetic Steklov operator on differential forms

In this paper, we introduce the magnetic Steklov operator on differential forms and show that the underlying boundary value problem is well-posed. Moreover, we show that an analogue of the Diamagnetic Inequality does not always hold for this operator, and we present some spectral computations of magnetic Steklov operators for $2$-dimensional and $4$-dimensional balls in Euclidean space.

math.SP↗

arXiv2025

Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

We consider the Steklov problem on differential $p$-forms defined by M. Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $0$-forms), highlighting the influence of the underlying topology on the spectrum for $p$-forms in general.

math.DG↗

arXiv2025

A note on the magnetic Steklov operator on functions

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean $2$-ball and $4$-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.

math.DG↗