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Register a new userThe Weighted Mean Curvature Derivative of a Space-Filling Diagram
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Representing an atom by a solid sphere in $3$-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [HRC13,RHK06] writes the latter as a linear combination of weight
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The morphometric approach [HRC13,RHK06] writes the solvation free energy as a linear combination of weight
In this article, we study hypersurfaces $Sigmasubset mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for self-shrinker. More specifically, they showed that a complete CWMC hypersurface with polynomial volume growth, bounded norm of the second fundamental form and that satisfies $|A|^2H(H-lambda)leq H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions if the reverse inequality holds then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete CWMC hypersurface with $Hgeq 0$ must either be a hyperplane or a generalized cylinder.
In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a constant weighted mean curvature hypersurface in the Gaussian shrinking Ricci soliton (not necessarily properly immersed) with a certain assumption on the integral of the second fundamental form must be a generalized cylinder.
In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $Sigmasubsetmathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.
Let $nge 2$ and $kge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $delta<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $lambda_1(M)$ of $M$ satisfies $$lambda_1(M)le nleft(delta +frac{1}{operatorname{Vol} M}int_M |H|^2 operatorname{dvol}right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an open problem raised by E. Heintze in 1988.
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