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In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R+ or R with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237-2274] on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative. [ABSTRACT FROM AUTHOR]

In this paper, we apply our previous estimates in Chen and Cheng [ On the constant scalar curvature Kähler metrics (I): a priori estimates , Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of K-energy in terms of L1 geodesic distance d1 in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the K-energy in (E1, d1) are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized L1 geodesic ray where the K-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs. [ABSTRACT FROM AUTHOR]

In this paper, we describe the flow of 2-surfaces in $ \mathbb{R}^{3}$ $ K^{-\alpha }$ $ \frac{1}{2}<\alpha \leq 1$ $ \alpha \in (0,\frac{1}{2}]$ by Schnürer. [ABSTRACT FROM AUTHOR]

In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject. [ABSTRACT FROM AUTHOR]