Simon Raulot
In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2.
A logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector
该研究在微分几何领域证明了非负截面曲率流形中具常长度平均曲率向量的闭子流形满足对数Sobolev不等式，丰富了子流形几何的分析工具。
We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.
We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $λ_0(g_0)>0$ to the case $λ_0(g_0)=0$.
$(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds
本文在${\rm Ric}_{\infty} \geq -K$条件下，基于全局$p$-Poincaré不等式推导出向前完备Finsler度量测度流形上的$(p, q)$-Sobolev不等式与Nash不等式，并建立了具有最佳常数的全局最优不等式。
Sobolev Inequality on Manifolds With Asymptotically Nonnegative Bakry-Émery Ricci Curvature
本文受已有工作启发，在具有密度且Bakry-Émery Ricci曲率渐近非负的流形上证明了Sobolev不等式。
We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$ Sobolev--Poincaré inequality developed by L.~Shartser.
Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay
本文利用ABP方法，在具有下二次曲率衰减的完备黎曼流形中，为紧致区域和子流形建立了Sobolev不等式。
In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local $ν$-functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.
This note is a continuation of [CMZ21]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy must also have uniformly bounded $ν$-functional. Consequently, on such an ancient solution there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [Bam20c] is valid, and in particular, when the underlying manifold is closed.
The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature
本文在具有渐近非负中间里奇曲率和欧氏体积增长的完备非紧黎曼流形上，证明了子流形的尖锐对数-Sobolev不等式，推广了Dong-Lin-Lu等人的已有结果。
The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields
本研究利用ABP方法，在非负截面曲率黎曼流形的紧子流形上，建立了光滑对称一致正定(0,2)张量场的Michael-Simon-Sobolev不等式，推广了S. Brendle的已有结果。
We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
A sharp Sobolev trace inequality of order four on three-balls
本文研究三球上的四阶尖锐 Sobolev 迹不等式，并揭示其与二维球面上三阶尖锐 Sobolev 不等式的等价性。
In this article we get a time-dependent Sobolev inequality along the Ricci flow which generalizes the earlier results of Zhang, Ye, Hsu. As an application of the time-dependent Sobolev inequality, we also get a growth of the ratio of bob-collapsing along the Ricci flow.
We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.
Proof of the Michael-Simon-Sobolev inequality using optimal transport
本文用最优输运方法重新证明了 Michael-Simon-Sobolev 不等式，并指出该不等式在余维数为 2 的子流形情形下达到最优。
Let M be a complete n-dimensional Riemannian manifold, if the sobolev inqualities hold on M, then the geodesic ball has maximal volume growth; if the Ricci curvature of M is nonnegative, and one of the general Sobolev inequalities holds on M, then M is diffeomorphic to $R^{n}$.
Logarithmic Sobolev inequality in manifolds with nonnegative curvature via the ABP method
本文通过ABP方法，在非负曲率流形上建立了最优的对数Sobolev不等式，并揭示了最优常数与渐近体积比的关系。
We give a new proof of Aubin's improvement of the Sobolev inequality on $\mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $\mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality.
A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the $L^2$ Folland-Stein embedding theorem is determined.
In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the mo…
The Logarithmic Sobolev inequality on non-compact self-shrinkers
本文利用 ABP 方法将 Brendle 关于闭自收缩子的最优对数 Sobolev 不等式推广到欧氏空间中完备、非紧、适当嵌入的自收缩子，得到 Ecker 结果的尖锐版本。
The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, we have proved that if the $Q'$-curvature is nonnegative, and the integral of $Q'$-curvature is below the dimensional bound $c_1'$, then we have the isoperimetric inequality. In this paper, we manage to drop the condition on the nonnegativity of the $Q'$-curvature. We prove that the volume form $e^{4u}$ is a strong $A_\infty$ weight. As a corollary, we prove the Sobolev-Poincaré inequality on a class of CR-manifolds with integrable $Q'$-curvature.
The sharp Sobolev inequality and its stability: An introduction
本讲义以尖锐Sobolev不等式及其分数阶推广为例，系统介绍最优函数不等式，重点讨论优化序列紧性、优化子刻画与定量稳定性。
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。