arXiv research

Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

We characterize how to vary the Abel-Jacobi map in terms of Schiffer variation. From this characterization, we will interpret the relation of hyperellipticity of curves with Schiffer variation and describe the deformation of elliptic solitons under Schiffer variation.

In this paper, we generalize the Hersch-Payne-Schiffer inequality for Steklov eigenvalues to higher dimensional case by extending the trick used by Hersch, Payne and Schiffer to higher dimensional manifolds.

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping t…

In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.

Deformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation ex…

Scattering theory for harmonic one-forms on Riemann surfaces.

We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. …

Study complex deformations of the circle using group cohomology and Virasoro algebra.

In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch-Payne-Schiffer , Brock}, Raulot-Savo and Dittmar.

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $Λ$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We inves…

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Her…

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infini…

We study the Hilbert manifold structure on $T_{0}(1)$ -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on $T_{0}(1)$ in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators $B_{1}$ and $B_{4}$, associated with the points in $T_{0}(1)$ via conformal w…

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

In this note, we use a result of Osserman and Schiffer \cite{OS} to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To the best of our knowledge, this fact has gone unremarked upon in the literature. …

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

Let $M$ be a Riemannian manifold and $Ω$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $Ω$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the nor…

Optimizes eigenvalues on surfaces with symmetries.

New subdomains found on cylinder and sphere with nonconstant curvatures.

The paper examines how isoparametric foliations affect the Pompeiu property in compact Riemannian manifolds.

We show uniqueness for overdetermined elliptic problems defined on topological disks $Ω$ with $C^2$ boundary, i.e., positive solutions $u$ to $Δu + f(u)=0$ in $Ω\subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte$ along $\partial Ω$, $\vecη$ the unit outward normal along $\partialΩ$ under the…

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known…

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Introduces a new elliptic operator with positive eigenvalue.

Proves Kato inequalities for various conformal operators.

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…

Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.

Local index theorem for chiral geometric operators proved using heat kernel.

Study estimates eigenvalues for concave Hessian operators on convex domains.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

GJMS operators connect geometry, analysis, and physics.

Extends Calabi operator to Riemannian locally symmetric spaces.

New spectral torsion defined for rescaled Dirac operators.

Proves formal self-adjointness of certain differential operators.

Researchers create new operators from Riemannian invariants.

The paper proves new theorems about specific types of operator perturbations.

Study of Dirac-like operators on spin manifolds with large mass parameters.

Study essential spectrum of differential operators on geometrically finite orbifolds.

The study proves inequalities for complex operators on curved spaces.

Study on opers over complex manifolds of dimension one.

Mixtures of neural operators reduce active complexity in operator learning.

Formula for Hadamard coefficients from Green's operators on spacetimes.

Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.

Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.