Niccolò Salvatori, Simon Scott
The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.
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We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure $φ$ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to $φ$ will converge to a torsion-free G_2 structure which is in the orbit of $φ$ under diffeomorphisms isotopic to the identity.
We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster's metric, pseudohermitian manifold.
On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+δ$ and $Δ_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+δ)Δ_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian…
In this paper we prove the Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of our identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb{R}^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded domains of $\mathbb{R}^n$, some sharp estimates for the torsion function of compact manifolds with boundary and a nonexistence result for the solutions of the Laplace equation on closed Riem…
We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed $G_2$-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free $G_2$-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsion-free $G_2$-structures. We also present a number of results which constitute a fairly complete algebraic and analytic basis for studying the Laplacian flow.
Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition
本文研究了Robin边界条件下拉普拉斯算子扭力和第一特征值与几何量的关系，证明了Robin扭力的上界及相关形状函数的最优值由滑动域实现。
Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
该研究证明有界凸区域中受限半拉普拉斯算子的扭转函数的二阶导数矩阵处处负定，通过反射调和延拓、连续性方法等完成论证，为相关偏微分方程理论提供关键结论。
The torsion function of a convex planar domain has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function on convex planar domains of high eccentricity. We obtain an approximation for the torsion function by viewing the domain as a perturbation of a rectangle in order to define an approximate Green's function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to compare the b…
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of $s$-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for $s\in(1,\sqrt{3}+3/2)$ in any dimension $n\geq 2$. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not sati…
Torsional Rigidity on Metric Graphs with Delta-Vertex Conditions
本文研究服从 δ-顶点条件的度量图 Laplace 算子，建立扭转刚度的变分刻画与 Hadamard 型公式，并据此给出界、极值图及正性判据。
Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.
Bounds are obtained for the $L^p$ norm of the torsion function $v_Ω$, i.e. the solution of $-Δv=1,\, v\in H_0^1(Ω),$ in terms of the Lebesgue measure of $Ω$ and the principal eigenvalue $λ_1(Ω)$ of the Dirichlet Laplacian acting in $L^2(Ω)$. We show that these bounds are sharp for $1\le p\le 2$.
In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analog of Yau's gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.
Joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds
本文研究三维Heisenberg Bieberbach流形上次拉普拉斯算子与Reeb向量场的联合谱，利用theta函数和有限群特征理论明确计算了其重数。
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.
Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations
本文在不假定度量束状结构和叶的总测地或最小条件的情况下，为一般黎曼流形的水平拉普拉斯算子发展了广义曲率维数不等式，并应用该框架得到多种几何和分析结果。
Let $Ω$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The fractional Cheeger constant $h_s (Ω)$, $0<s<1$, is defined by \[h_s(Ω)=\inf_{E\subsetΩ}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N }\frac{|χ_{E}(x)-χ_{E}(y)|}{|x-y|^{N+s}} dx dy,\] with $χ_{E}$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that \[\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},\] where $ϕ_p^s$ is the fractional $(s,p)$-torsio…
Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\varepsilon\downarrow 0$ of the $L^{\infty}$ norm of the torsion function with a Neumann boundary condition on $\partial D$, and a Dirichlet boundary condition on $\partial (\varepsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Mars…
Let (M,g) be an odd-dimensional incomplete compact Riemannian singular space with a simple edge singularity. We study the analytic torsion on M, and in particular consider how it depends on the metric g. If g is an admissible edge metric, we prove that the torsion zeta function is holomorphic near s = 0, hence the torsion is well-defined, but possibly depends on g. In general dimensions, we prove that the analytic torsion depends only on the asymptotic structure of g near the singular stratum of M; when the dimension of the edge is odd, we prove that the analytic torsion is independent of the …
Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp.\ any) non--integer order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace w…
The Cheeger problem for a bounded domain $Ω\subset\mathbb{R}^{N}$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subsetΩ$ and the Cheeger constant $h(Ω)$ is the minimum of these quotients. Let $ϕ_{p}\in C^{1,α}(\barΩ)$ be the $p$-torsion function, that is, the solution of torsional creep problem $-Δ_{p}ϕ_{p}=1$ in $Ω$, $ϕ_{p}=0$ on $\partialΩ$, where $Δ_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that $\lim_{p\rightarrow1^{+}}(\|ϕ_{p}…
On efficiency and localisation for the torsion function
本文研究Dirichlet拉普拉斯和Schrödinger算子扭转函数的效率与局部化，并探讨其与第一Dirichlet特征函数几何的关联。
For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\Om \subset \R^m$ satisfies formally the equation $-Δ_p =1$ in $\Om$ and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on $\partial \Om$. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corr…
Witten deformation for non-Morse functions and gluing formula for analytic torsions
本文针对非 Morse 函数族提出 Witten 变形的新解析方法，证明解析挠率的粘合公式，并揭示其与 Bismut-Zhang 定理及移动边界问题的联系，为高阶推广奠定基础。
Let $Ω$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_Ω$ denote the torsion function for $Ω$. It is known that $v_Ω$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(Ω)$, denoted by $λ(Ω)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_Ω\|_{\Leb^{\infty}(Ω)}λ(Ω)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $ε>0$ we construct an open, bounded and connected set $Ω_ε\subset \R^m$ such that $\|v_{Ω_ε}\|_{\Leb^{\infty}(Ω_ε)} λ(Ω_ε)<1+ε$. An upper bound for $v_Ω$ is obtained for planar, convex sets…
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