Related papers: A split special Lagrangian calibration associated …

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates.…

In this paper, we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding…

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges.…

For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$…

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of…

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with…

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply…

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving…

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector rather than the velocity. It enables a feasible Lagrangian study of fluid dynamics, as the isosurface of…

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise…

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic…

We study the interplay between the global causal and geometric structures of a spacetime $(M,g)$ and the features of a given smooth $\mathbb{R}$-action $\rho$ on $M$ whose orbits are all causal curves, building on classic results about Lie…

In this paper we study the rigidity problem for sub-static systems with possibly non-empty boundary. First, we get local and global splitting theorems by assuming the existence of suitable compact minimal hypersurfaces, complementing recent…

This paper deals with Riemannian optimization on the unit sphere in terms of $p$-norm with general $p > 1$. As a Riemannian submanifold of the Euclidean space, the geometry of the sphere with $p$-norm is investigated, and several geometric…

We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence…

We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number…

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation…

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact…

Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative…