arXiv subjects
Publications and source records attributed to Sugata Mondal.
We extend the Otal-Rosas bound on the number of small eigenvalues of the Laplacian on a hyperbolic surface to the small eigenvalues of pseudo-Laplacians. In the process, we extend the work of Colin de Verdière on the spectral theory of pseudo-Laplacians to hyperbolic surfaces with more than one cusp.
We prove the non-existence of new eigenvalues in $[0,Λ]$ for specific and random finite coverings of a complete and connected Riemannian manifold $M$ with Ricci curvature bounded from below, where $Λ$ is any positive number below the essential spectrum of $M$ and the spectrum of the universal cover of $M$, provided the representation theory of the fundamental group of $M$ satisfies certain conditions.
We study the behaviour of eigenvalues, below the bottom of the essential spectrum, of the Laplacian under finite Riemannian coverings of complete and connected Riemannian manifolds. We define spectral stability and instability of such coverings. Among others, we provide necessary conditions for stability or, equivalently, sufficient conditions for instability.
Let $Ω$ be a bounded centrally symmetric domain in $\mathbb{R}^2$ with analytic boundary $\partial Ω$ and center $c$. Let $τ= τ(Ω)$ be the number of points $p$ on $\partial Ω$ such that the normal line to $\partial Ω$ at $p$ passes through $c$. We show that if $τ< 8$ then $Ω$ satisfies the Schiffer's conjecture.
We study the set of critical points of a solution to $Δu = λ\cdot u$ and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon $P$ has infinitely many critical points, then $P$ is a rectangle.
We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.
We show that a second Neumann eigenfunction u of a Euclidean triangle has no critical points in the interior of the triangle.
Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.
In our previous work we introduced, for a Riemannian surface $S$, the quantity $ Λ(S):=\inf_Fλ_0(F)$, where $λ_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $Λ(S)\geλ_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $Λ(S)$ with the systole, improving a result by the last named author.
We discuss our recent work on small eigenvalues of surfaces. As an introduction, we present and extend some of the by now classical work of Buser and Randol and explain novel ideas from articles of Sévennec, Otal, and Otal-Rosas which are of importance in our line of thought.
For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of πin A(S).
The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked length spectrum. The purpose of this paper is to introduce a closely a related unmarked spectrum, the length-angle spectrum, and show that it determines the surface uniquely.
Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and negative Euler characteristic has at most negative of the Euler characteristics many small eigenvalues.
We apply topological methods to study the smallest non-zero number $λ_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {λ_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.
We show that the Laplacian of a Riemannian metric on a closed surface S with Euler characteristic χ(S) < 0 has at most -χ(S) small eigenvalues.
Let $S$ be a noncompact, finite area hyperbolic surface of type $(g, n)$. Let $Δ_S$ denote the Laplace operator on $S$. As $S$ varies over the {\it moduli space} ${\mathcal{M}_{g, n}}$ of finite area hyperbolic surfaces of type $(g, n)$, we study, adapting methods of Lizhen Ji \cite{Ji} and Scott Wolpert \cite{Wo}, the behavior of {\it small cuspidal eigenpairs} of $Δ_S$. In Theorem 2 we describe limiting behavior of these eigenpairs on surfaces ${S_m} \in {\mathcal{M}_{g, n}}$ when $({S_m})$ converges to a point in $\overline{\mathcal{M}_{g, n}}$. Then we consider the $i$-th {\it cuspidal eigenvalue}, ${λ^c_i}(S)$, of $S \in {\mathcal{M}_{g, n}}$. Since {\it non-cuspidal} eigenfunctions ({\it residual eigenfunctions} or {\it generalized eigenfunctions}) may converge to cuspidal eigenfunctions, it is not known if ${λ^c_i}(S)$ is a continuous function. However, applying Theorem 2 we prove that, for all $k \geq 2g-2$, the sets $${{\mathcal{C}_{g, n}^{\frac{1}{4}}}}(k)= \{ S \in {\mathcal{M}_{g, n}}: {λ_k^c}(S) > \frac{1}{4} \}$$ are open and contain a neighborhood of ${\cup_{i=1}^n}{\mathcal{M}_{0, 3}} \cup {\mathcal{M}_{g-1, 2}}$ in $\overline{\mathcal{M}_{g, n}}$. Moreover, using topological properties of nodal sets of {\it small eigenfunctions} from \cite{O}, we show that ${{\mathcal{C}_{g, n}^{\frac{1}{4}}}}(2g-1)$ contains a neighborhood of ${\mathcal{M}_{0, n+1}} \cup {\mathcal{M}_{g, 1}}$ in $\overline{\mathcal{M}_{g, n}}$. These results provide evidence in support of a conjecture of Otal-Rosas \cite{O-R}.
We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${λ_{2g-2}}(S)$: ${λ_{2g-2}}(S) > 1/4 + {ε_0}(S)$, where ${ε_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$