We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.
Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-Δ$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\nabla^*\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bound $<1$, if the function $\maxσ(V^-)$ is in the Kato class of $(M,g)$. In particular, this gives a sufficient condition under which one can define the form sum $H(V):=H(0)\dotplus V$ on arbitrary Riemannian manifolds.
We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-type operator on smooth functions on closed Riemannian manifolds. We then apply the result to establish some new domination inequalities for spectral zeta functions and their related spectral zeta kernels on $n$-dimensional unit spheres using Kato's inequalities and majorisation techniques. Our results are the generalisations of Kato's comparison inequalities for Riemannian surfaces to $n$-dimensional closed Riemannian manifolds.
By introducing the concept of \emph{Kato control pairs} for a given Riemannian minimal heat kernel, we prove that on every Riemannian manifold $(M,g)$ the Kato class $\mathcal{K}(M,g)$ has a subspace of the form $\mathsf{L}^q(M,d\varrho)$, where $\varrho$ has a continuous density with respect to the volume measure $μ_g$ (where $q$ depends on $\dim(M)$). Using a local parabolic $\mathsf{L}^1$-mean value inequality, we prove the existence of such densities for every Riemannian manifold, which in particular implies $\mathsf{L}^q_{loc}(M)\subset\mathcal{K}_{loc}(M,g)$. Based on previously establis…
We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti number estimates for closed manifolds whose Ricci curvature is controlled in the Kato class.
In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.
We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < ζ^x\}}\big] < \infty, \end{equation*} then the lifetime $ζ^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in …
In this paper, we extend the Hijazi type inequality, involving the Energy-Momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spin$^c$ manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.
We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the $L^p$-Calderón-Zygmund inequalities extend to a reasonable class of non-compact Riemannian manifolds without the assumption of a positive injectivity radius. In the present paper, we give a positive answer for $1<p<2$ under the natural assumption of a lower bound on the Ricci curvature. For $p>2$, we complement the study in Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality by adding Kato class bounds on the R…
In this paper we prove that a complete Riemannian manifold is $L^p$-positivity preserving for any $p\in(1,\infty)$. This means that any $L^p$ function which solves $(-Δ+ 1)u\ge 0$ in the sense of distributions is necessarily non-negative. In particular, the case $p=2$ of our result answers in the affermative a conjecture formulated by M. Braverman, O. Milatovic and M. Shubin in 2002. The two main ingredients are a new a-priori regularity result for positive subharmonic distributions, which in turn permits to prove a Liouville type theorem, and a Brezis-Kato inequality on Riemannian manifolds. …
We consider the $L^2$ gradient flow for the Willmore functional in Riemannian manifolds of bounded geometry. In the euclidean case E.\;Kuwert and R.\;Schätzle [\textsl{Gradient flow for the Willmore functional,} Comm. Anal. Geom., 10: 307-339, 2002] established a lower bound of a smooth solution of such a flow, which depends only on how much the curvature of the initial surface is concentrated in space. In a second joint work [\textsl{The Willmore flow with small initial energy,} J.\;Differential Geom., 57: 409-441, 2001] the aforementioned authors proved that a suitable blow-up converges to a…
We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the curve of eigenvalues, which is used as a device to prove that the eigenvalues of the Laplace-Neumann operator are generically simple in the space $\mathcal{M}^k$ of all $C^k$ Riemannian metrics on $M$. This implies the existence of a residual set of metrics in $\mathcal{M}^k$, which…
Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform our analysis on $(Ω,g,μ)$. The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $μ$ is unconditional, i.e. invariant under reflection with respect to the principle hyperpla…
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics' in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricc…
Let $Ω\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schrödinger operator $-Δ+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $Ω$ is the statement $$ \text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus Ω$ one has $f|_Ω\in W^{1,2}_{0}(Ω;V)$} $$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $Ω$ is Kac regular, a probabilistic property w…
