Related papers: Quantitative marked length spectrum rigidity for s…

Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f\circ h$ for $f\in…

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on…

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space…

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric…

Let $\gamma$ be a filling curve on a topological surface $\Sigma$ of genus $g \geq 2$. The inf invariant of $\gamma$, denoted $m_{\gamma}$, is the infimum of the length function on the space of marked hyperbolic structures on $\Sigma$. This…

Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a…

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the…

Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This extends…

By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip-Fisher-Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively $s$-curved magnetic system on a closed…

We prove that among the set of smooth diffeomorphisms there exists a $C^1$-open and dense subset of data such that either the Lagrange spectrum is finite and the dynamics is a Morse-Smale diffeomorphism or the Lagrange spectrum has positive…

Let $M$ be a closed smooth Riemannian manifold $M$, and let $f:M\to M$ be a diffeomorphism. Herein, we demonstrate that (i) if $f$ has the $C^1$ robustly inverse shadowing property on the chain recurrent set $\mathcal{CR}(f)$, then…

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to…

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant…

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of…

For warped products with harmonic curvature, nonconstant warping functions $\phi$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=\phi^{-2}h$ is constant and negative,…

Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to $\mathbb{S}^{n-1}(1)$. We prove some rigidity…

In this paper, we prove that if the area functional of a surface $\Sigma^2$ in a symplectic manifold $(M^{2n},\bar{\omega})$ has a critical point or has a compatible stable point in the same cohomology class, then it must be…

We find sharp upper bounds for the multiplicities and the numerical values of all the distinct eigenvalues on a surface of revolution diffeomorphic to the sphere.

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a…

We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in…