相关论文: Minimal area of Finsler disks with minimizing geod…

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}{\pi}R^2$. This inequality has long been conjectured for $R$ less…

By refining the volume estimate of Heintze and Karcher \cite{HK}, we obtain a sharp pinching estimate for the genus of a surface in $\mathbb S^{3}$, which involves an integral of the norm of its traceless second fundamental form. More…

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It…

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his…

Motivated in part by the bi-gravity approach to massive gravity, we introduce and study the multimetric Finsler geometry. For the case of an arbitrary number of dimensions, we study some general properties of the geometry in terms of its…

In 1974, Federer proved that all area-minimizing hypersurfaces on orientable manifolds were calibrated by weakly closed differential forms. However, in this manuscript, we prove the contrary in higher codimensions: calibrated…

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type…

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed…

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all…

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual…

In this short note, we prove that if $F$ is a weak upper semicontinuous admissible Finsler structure on a domain in $\mathbb{R}^n$, $n\geq 2$, then the intrinsic distance and differential structures coincide.

Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(\Omega)$ into Orlicz spaces for an arbitrary domain $\Omega\subset \mathbb{R}^n$ with…

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under…

Let $B_2(p)$ be an $n$-dimensional smooth geodesic ball with Ricci curvature $\geq-(n-1)\kappa^2$ for some $\kappa\geq0$. We establish the Sobolev inequality and the uniform Neumann-Poincar\'e inequality on each minimal graph over $B_1(p)$…

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max…

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with finite fundamental group and $n\ge 2$. Thus generically there are at least two closed geodesics on compact Finsler manifolds with…

We establish a framework, namely, nuclear bounded Fr\'{e}chet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed…

In [dLMu05], DeLellis and M\"uller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ \Sigma \subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(\Sigma) = 4 \pi\ $, it was shown that $\…

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension…

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of…