arXiv subjects
Publications and source records attributed to Emmanuel Humbert.
The constraint equations of general relativity can in many cases be solved by the conformal method. We show that a slight modification of the equations of the conformal method admits no solution for a broad range of parameters. This suggests that the question of existence or non-existence of solutions to the original equations is more subtle than could perhaps be expected.
This text aims to explains briefly the formalism of quantum mechanics to mathematicians.
We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^p for p=2n/(n-2) are also L^2. This L^p-L^2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As an application we see that the smooth Yamabe invariant of any 2-connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions at least 11.
Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.
Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $σ\in W^{1,p}$ and a positive function $τ\in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $σ$ and $τ$. We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on $1$-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics $g$ for the $C^0$-topology.
Let $g$ be a metric on $S^3$ with positive Yamabe constant. When blowing up $g$ at two points, a scalar flat manifold with two asymptotically flat ends is produced and this manifold will have compact minimal surfaces. We introduce the $\Th$-invariant for $g$ which is an isoperimetric constant for the cylindrical domain inside the outermost minimal surface of the blown-up metric. Further we find relations between $\Th$ and the Yamabe constant and the existence of horizons in the blown-up metric on $\mR^3$.
We show a surgery formula for the relative Yamabe invariant and give applications to the study of concordance classes of metrics.
We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $σ$-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least 2, then $τ(N) \geq \min\{τ(M),Λ_n\}$ with $Λ_n>0$. Various topological conclusions can be drawn, in particular that τis a spin-bordism invariant below $Λ_n$. Below $Λ_n$, the values of $τ$ cannot accumulate from above when varied over all manifolds of a fixed dimension.
We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert.
Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten, the constant term in the asymptotic development of the Green's function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. In the present article, we will give a proof for this fact which is considerably shorter than previous proofs. Our proof is a modification of Witten's argument, but no analysis on asymtotically flat spaces is needed.
Let $(M,g,\si)$ be a compact spin manifold of dimension $n \geq 2$. Let $λ_1^+(\tilde{g})$ be the smallest positive eigenvalue of the Dirac operator in the metric $\tilde{g} \in [g]$ conformal to $g$. We then define $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. We show that $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %=\frac{n}{2} \om_n^{1 \over n}$ . We find sufficient conditions for which we obtain strict inequality $\lamin(M,[g],\si) < \lamin(\mS^n)$. This strict inequality has applications to conformal spin geometry. ----- Soit $(M,g,\si)$ une variété spinorielle compacte de dimension $n \geq 2$. %Si $\tilde{g} \in [g]$ est une métrique conforme à $g$, On note $λ_1^+(\tilde{g})$ la plus petite valeur propre $>0$ de l'opérateur de Dirac dans la métrique $\tilde{g} \in [g]$ conforme à $g$. On définit $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. On montre que $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %= \frac{n}{2} \om_n^{1 \over n}$ On trouve des conditions suffisantes pour lesquelles on obtient l'inégalité stricte $\lamin(M,[g],\si) < \lamin(\mS^n)$. Cette inégalité stricte a des applications en géométrie spinorielle conforme.
Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case $n \geq 3$ and in the case $n = 2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\neq \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2\tildeλ^2\leq \tildeμ$, where $\tildeμ$ denotes the first positive eigenvalue of the Laplace operator.
We give geometrical conditions under which there exist extremal functions for the sharp $L^2$-Nash inequality.
The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω\subset Ω$ decays exponentially in time as soon as $ω$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $τ(ω)$ of this decay satisfies $τ(ω)= 2 \min(-μ(ω), g(ω))$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $μ(ω)$ denotes the spectral abscissa of the damped wave equation operator and $g(ω)$ is a number called the geometrical quantity of $ω$ and defined as follows. A ray in $Ω$ is the trajectory generated by the free motion of a mass-point in $Ω$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(ω)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(ω)$ when $ω$ is a finite union of squares.
Let $(M,g)$ be a compact Riemannian manifold on dimension $n \geq 4$ not conformally diffeomorphic to the sphere $S^n$. We prove that a smooth function $f$ on $M$ is a critical function for a metric $\tilde{g}$ conformal to $g$ if and only if there exists $x \in M$ such that $f(x)>0$.
Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $λ(M,[g],\si)= \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $λ_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that $λ(M,[g],\si) \leq λ(\mS^n) =\frac{n}{2} \om_n^{1/n}$ where \om_n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n \geq 2 such that D is invertible. E.g. we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.
Let M be a compact manifold with a spin structure χand a Riemannian metric g. Let λ_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M,χ):= sup inf \sqrt{λ_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that τ(T^2,χ)=2\sqrtπ if χis ``the'' non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2\sqrtπ at one end of the spin-conformal moduli space.
Let $(M,g,σ)$ be a compact Riemmannian surface equipped with a spin structure $σ$. For any metric $\tilde{g}$ on $M$, we denote by $μ\_1(\tilde{g})$ (resp. $λ\_1(\tilde{g})$) the first positive eigenvalue of the Laplacian (resp. the Dirac operator) with respect to the metric $\tilde{g}$. In this paper, we show that $$\inf \frac{λ\_1(\tilde{g})^2}{μ\_1(\tilde{g})} \leqslant {1/2}.$$ where the infimum is taken over the metrics $\tilde{g}$ conformal to $g$. This answer a question asked by Agricola, Ammann and Friedrich