Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincaré-Einstein metrics. The results are formulated in terms of Weyl conformal invariants arising from the ambient metric of Fefferman-Graham. As applications we obtain "Green function" characterizations of locally conformally flat manifolds and a spectral theoretic characterization of the conformal class of the round sphere.
In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the $ε$-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is $O\left(\left(\log n/n\right)^{1/(m+2)}\right)$, where $m$ and $n$ denote the dimension of the manifold and the sample size, respectively.
We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the …
Using Roelcke formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $κ$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.
We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-Müller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.
I prove the existence of small time heat expansion for the Laplace operator on an analytic hypersurface with an isolated singularity. First we obtain a local parametrization of the hypersurface near the singularity. We introduce the notion of quasihomogeneous tangent cone. Then perturb the parametrization of the cone employing a Newton scheme and obtain a parametrization with functions of specific form. These allow us to obtain local models for the Laplace operator near the singularity. These are operators with irregular singularities. We derive the estimates required by singular asymptotics.
Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4π$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $ζ$-regularized determina…
We construct radial fundamental solutions for the differential form Laplacian on negatively curved symmetric spaces. At least one of these Green's functions also yields a Biot-Savart Opearator, i.e. a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential. Any Biot-Savart operator gives rise to a Gauss linking integral.
In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally harmonic manifolds.
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our appr…
We study the classification problem of singularities of function-germs with harmonic leading terms of two variables under the right-equivalence. We observe that the multiple actions of Laplacian appear for the classifications of such class of function-germs.
We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.
This is the first step in an attempt at a deformation theory for $G_{2}-$instantons with $1-$dimensional conic singularities. Under a set of model data, the linearization yields a self-adjoint first order elliptic operator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimension reduction, the operator $P$ also arises from Hermitian Yang-Mills connections with isolated conic singularities on a Calabi-Yau $3$-fold. Using the Quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$ (denoted by $Spec P$). We show that $SpecP$ consists o…
