In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)κ$, $\, κ\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian.
A formal treatment of Killing 1-form and 2-Killing 1-form on Riemannian Poisson manifold, Riemannian Poisson warped product space are presented. In this way, we obtain Bochner type result on compact Riemannian Poisson manifold, compact Riemannian Poisson warped product space for Killing 1-form and 2-Killing 1-form. Finally, we give the characterization of a 2-Killing 1-form on $(\mathbb{R}^2,g,Π)$.
In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner's identity, and leads to "semilinear Calderón-Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclide…
On the Riemannian Geometry of tangent Poisson-Lie group
设$(G,Π_{G},\tilde{g})$是一个Poisson-李群，它具有一个左不变的伪黎曼度量$\tilde{g}$，$(TG,Π_{TG},\tilde{g}^{c})$是$G$的Sanchez de Alvarez正切Poisson-李群，它具有一个左不变的伪黎曼度量$\tilde{g}^{c}$，是$\tilde{g}$的完全提升.本文用Poisson-李群$(G,Π_{G},\tilde{g})$的Levi-Civita联络、曲率和亚曲率分别表示$(TG,Π_{TG},\tilde{g}^{c})$的Levi-Civita联络、曲率和亚曲率，证明了$G$上的微分形式空间$Ω^{*}(G)$是微分分次Poisson代数当且仅当$Ω^{*}(TG)$是微分分次Poisson代数.证明了三元组$(G,Π_{G},\tilde{g})$是伪黎曼Poisson-李群当且仅当$(TG,Π_{TG},\tilde{g}^{c})$也是伪黎曼Poisson-李群，并给出了一个6维伪黎曼Sanchez de Avarez正切Poisson-李群的例子.
In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.
We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.
We show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type.
A smooth curve on $G/H$ is called a Riemannian equigeodesic if it is a homogeneous geodesic for all $G$-invariant Riemannian metrics on $G/H$. With the $G$-invariant Riemannian metric replaced by other classes of $G$-invariant metrics, we can similarly define Finsler equigeodesic, Randers equigeodesic, $(α,β)$ equigeodesic, etc. In this paper, we study Randers and $(α,β)$ equigeodesics. For a compact homogeneous manifold, we prove Randers and $(α,β)$ equigeodesics are equivalent, and find a criterion for them. Using this criterion we can classify the equigeodesics on many compact homogeneous m…
We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full isometry group of $M$. Namely, we prove that there exists a transvection of $M$ in the direction of any element of the nullity, possibly by enlarging the presentation group $G$. Moreover, we prove that these transvections generate an abelian ideal of $\tilde{\mathfrak g}$. These results constitute a substantial improvement on the structure theory developed in \cite{DOV}. …
