A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds
Guofang Wei, Rugang Ye
备注：
In Theorem A, the previous maximum estimate in terms of the isoperimetric constant is replaced by a maximum estimate in terms of the volume-normalized isoperimetric constant. The statements of Gallot's estimate for the isoperimetric constant are corrected
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curvature bounded from below and diameter bounded from above to yield a maximum estimate without dependence on a positive lower bound for the volume.
We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the curve of eigenvalues, which is used as a device to prove that the eigenvalues of the Laplace-Neumann operator are generically simple in the space $\mathcal{M}^k$ of all $C^k$ Riemannian metrics on $M$. This implies the existence of a residual set of metrics in $\mathcal{M}^k$, which…
In this paper, we consider the global regularity for Monge-Ampère type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained under some necessary assumptions. Our main result extends the main theorem from the case of Euclidean space $R^n$ in [11] to Riemannian manifolds.
Capacities-to-masses for Laplace-Beltrami operators on complete Riemannian manifolds
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Mathematics & Statistics, Memorial University（纪念大学数学与统计系）
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A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable …
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