This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.

In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound

$$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$

$d$

${\mathcal{M}}$

$V_{{\rm\Omega}}(x,r)$

$B(x,r)\cap {\rm\Omega}$

$B(x,r)$

$x$

$r$

${\it\delta}$

${\rm\Omega}$

$e^{-t{\mathcal{A}}}$

$L^{p}({\rm\Omega})$

$p\in [1,\infty )$

A Riemannian manifold $\left( M,\,\rho \right)$ is called Einstein if the metric $\rho $ satisfies the condition $\text{Ric}\left( \rho \right)\,=\,c\,\cdot \,\rho $ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO\left( n \right)/SO\left( l \right)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$$SO\left( n \right)$-invariant Einstein metrics.

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature KM [ges ] 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n([ges ]3)-dimensional complete connected Riemannian manifold with KM [ges ] 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.