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The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike…
The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity…
We present a differential geometric formulation of the Poincare problem using the calculus of moving surfaces (CMS). In this framework, an n dimensional compact hypersurface evolves under a velocity field that couples motion to the…
We derive formulas for the mean curvature of special Lagrangian 3-folds in the general case where the ambient 6-manifold has intrinsic torsion. Consequently, we are able to characterize those SU(3)-structures for which every special…
We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling…
We demonstrate the existence of branched immersed 2-spheres with prescribed mean curvature, with controlled Morse index and with arbitrary codimensions in closed Riemannian manifold $N$ admitting finite fundamental group, where $\pi_k(N)…
We consider the physics of a matrix model describing D0-brane dynamics in the presence of an RR flux background. Non-commuting spaces arise as generic soltions to this matrix model, among which fuzzy spheres have been studied extensively as…
We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical…
We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation…
We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…
Based on a novel type of Sobolev-Poincar\'e inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of…
We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce…
Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with…
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic…
We showed the existence of non-radial solutions of the equation $\Delta u -\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $\lambda$. We show that for any…
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work…
Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. We show that for a $C^\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence…
We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the…
Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally…