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All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat…

Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $\partial M$and $\partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of…

We study stable compact constant mean curvature surfaces in the product spaces S2 X R and H2 X R and in some other Riemannian 3-manifolds.

In this survey article we review several results on the curvature of semi-Riemannian metrics which are motivated by the positive mass theorem. The main themes are estimates of the Riemann tensor of an asymptotically flat manifold and the…

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the…

For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on…

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded…

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of…

Canonical metrics and conformal invariants are presented for closed oriented even-dimensional manifolds with non-degenerate conformal structures and in particular for compact Riemann surfaces.

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…

In this article, we consider a closed rank one $C^\infty$ Riemannian manifold $M$ of nonpositive curvature and its universal cover $X$. Let $b_t(x)$ be the Riemannian volume of the ball of radius $t>0$ around $x\in X$, and $h$ the…

For a Riemannian manifold $M$, possibly with boundary, we consider the Riemannian product $M\times\mathbb{R}^k$ with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with…

Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some…

In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the…

In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change…

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$…

We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of…

We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the…

We consider the inverse problem of determining the metric-measure structure of collapsing manifolds from local measurements of spectral data. In the part I of the paper, we proved the uniqueness of the inverse problem and a continuity…

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic…