Related papers: Mass endomorphism and spinorial Yamabe type proble…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in…

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a…

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $\Sigma$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean…

We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further…

We consider the problem of distortion minimal morphing of $n$-dimensional compact connected oriented smooth manifolds without boundary embedded in $\R^{n+1}$. Distortion involves bending and stretching. In this paper, minimal distortion…

In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$…

Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a…

Let $n\ge 2$ and $k\ge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than…

We bound the locations of outermost minimal surfaces in geometrostatic manifolds whose ADM mass is small relative to the separation between the black holes and prove the Intrinsic Flat Stability of the Positive Mass Theorem in this setting.

We consider the problem of minimizing the second conformal eigenvalue of the conformal Laplacian in a conformal class of metrics with renormalized volume. We prove, in dimensions $n\in\left\{3,\dotsc,10\right\}$, that a minimizer for this…

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension…

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except dimension 5 (n = 2). In…

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a…

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean…

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that was establishedby McCormick. First, we show that any asymptotically flat static…

In this article, we prove that on any compact spin manifold of dimension m congruent 0,6,7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by…

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative…

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra of type $D$ or $E$ and $\lambda$ be a dominant integral weight whose support bounds the subdiagram of type $D_4$. We study certain quantum affinizations of the simple $\mathfrak…

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number…