Related papers: Four closed characteristics on compact star-shaped…

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$ equipped with a generic Riemannian metric $g$. Let $c$ be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean…

Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where…

For the simply connected compact exceptional Lie group $E_8$, we determine the structure of subgroup $(E_8)^{\sigma, \sigma'}$ of $E_8$ which is the intersection $(E_8)^\sigma \cap (E_8)^{\sigma'}$. Then the space $E_8/(E_8)^{\sigma,…

The authors study the geometry of lightlike hypersurfaces on a four-dimensional manifold $(M, c)$ endowed with a pseudoconformal structure $c = CO (2, 2)$. They prove that a lightlike hypersurface $V \subset (M, c)$ bears a foliation formed…

In this paper we classify non-symplectic automorphisms of order 8 on complex K3 surfaces in case that the fourth power of the automorphism has only rational curves in its fixed locus. We show that the fixed locus is the disjoint union of a…

We study periodic orbits on a nondegenerate dynamically convex starshaped hypersurface in $\mathbb{R}^{2n}$ along the lines of Long and Zhu, but using properties of the $S^1$-equivariant symplectic homology. We prove that there exist at…

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…

Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within…

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits…

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal…

Several uniqueness results on compact maximal hypersurfaces in a wide class of sta- bly causal spacetimes are given. They are obtained from the study of a distinguished function on the maximal hypersurface, under suitable natural first…

We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.

There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains…

We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms $N^n(\rho)$ as $p$-elastic curves, for a suitable rational number $p\in[1/4,1)$ which depends on the dimension $n$ of the ambient space.…

We classify curvature homogeneous hypersurfaces in S^4 and H^4. In higher dimesnsion one only has the FKM examples and an isolate one by Tsukada of a hypersurface in H^5. Besides some simple examples, we show that there exists an isolated…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly…

In this paper, we found non-symplectic index of all supersingular K3 surfaces defined over a field of characteristic p>3.

We study germs of J-Holomorphic curves contained in $M$, a real analytic hypersurface of an symplectic manifold of dimension 4. We show, under topological hypothesis on $M$, that if $M$ is compact then $M$ is of finite type and so there is…

We consider the CSLs of 4-dimensional hypercubic lattices. In particular, we derive the coincidence index $\Sigma$ and calculate the number of different CSLs as well as the number of inequivalent CSLs for a given $\Sigma$. The hypercubic…

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an…