Related papers: Blow-up estimates at horizontal points and applica…

Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to…

We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of $(\bullet,0)$- and…

Given a random sample from a density function supported on a manifold $M$, a new method for the estimating highest density regions of the underlying population is introduced. The new proposal is based on the empirical version of the opening…

In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimension,…

Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard…

Many geometry processing pipelines implicitly assume their input data is a manifold, or is sampled from one, with a unique tangent plane at every point. Geometric data, however, routinely contains sharp features like edges, corners,…

We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstadt. Our proof does not rely on her proof. In addition, we study global isometries of general homogeneous manifolds equipped…

We show that the global and local constructions of three types of blowup of a smooth manifold along a closed submanifold in differential topology are equivalent.

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in…

We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique…

Let $f$ be a positive smooth function on a close Riemann surface (M,g). The $f-energy$ of a map $u$ from $M$ to a Riemannian manifold $(N,h)$ is defined as $$E_f(u)=\int_Mf|\nabla u|^2dV_g.$$ In this paper, we will study the blow-up…

In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff…

We study mean field equations with singular sources on a compact Riemann surface with boundary $(\Sigma,g)$, subject to homogeneous Neumann boundary conditions: \[ -\Delta_g v = \rho\left( \frac{V e^{v}}{\int_\Sigma V e^{v}\, d v_g} -…

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal…

We establish an area-type formula for the intrinsic spherical Hausdorff measure of every regular curve embedded in an arbitrary graded group.

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with…

We propose new tools based on basic lattice theory to calculate the integral cohomology of the quotient of a manifold by an automorphism group of prime order. As examples of applications, we provide the Beauville--Bogomolov forms of some…

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the…

In this paper, we apply a blow-up method of Schoen and Yau in \cite{SY81} to study a large class of prescribed mean curvature (PMC) Dirichlet problems in $n(n\geq 2)$-dimensional Riemannian manifolds. In this process we establish curvature…

In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider…