Related papers: On the Dual Geometry of Laplacian Eigenfunctions

In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…

In three dimensions, an abelian gauge field is related by duality to a free, periodic scalar field. Though usually considered on Euclidean space, this duality can be extended to a general three-manifold M, in which case topological features…

As key subjects in spectral geometry and combinatorial graph theory respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in…

To study the regularity of heat flow, Lin-Wang[1] introduced the quasi-harmonic sphere, which is a harmonic map from $M=(\mathbb{R}^m,e^{-\frac{|x|^2}{2(m-2)}}ds_0^2)$ to $N$ with finite energy. Here $ds_0^2$ is Euclidean metric in…

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to…

Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues…

By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(\tau)^2= F(1/2,1/2,1;\lambda(\tau))$, where…

Filinski constructed a symmetric lambda-calculus consisting of expressions and continuations which are symmetric, and functions which have duality. In his calculus, functions can be encoded to expressions and continuations using primitive…

We apply the techniques of our previous paper to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of…

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ \phi_\lambda $ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of…

Let $G$ a locally compact group and $(\Phi,\Psi)$ be a complementary pair of Young functions. Let $A_\Phi(G)$ be the Orlicz analogue of the classical Fig\`{a}-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and…

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $\Omega\subset K$ whose boundary $\partial \Omega$ is a graph-directed self-similar set, we prove that…

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function.…

We study globally defined $(\lambda,\mu)$-eigenfamilies on closed Riemannian manifolds. Among others, we provide (non-)existence results for such eigenfamilies, examine their topological properties and classify $(\lambda,\mu)$-eigenfamilies…

We introduce and analyze a Spencer-type elliptic complex on the space of differential forms valued in symmetric powers of an adjoint bundle, $\Omega^\bullet(X)\otimes \mathrm{Sym}^\bullet(G)$. The complex is governed by a total differential…

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed…

In this paper we numerically solve the eigenvalue problem $\Delta u + \lambda u = 0$ on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is…

We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the…

We consider arbitrary embeddings of surface operators in a pure, non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For any surface operator with a priori simultaneously non-vanishing parameters, we explicitly show…