We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken (J Reine Angew Math 382:35–48, 1987) who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the

-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.

We prove some basic properties of the relative Nakayama–Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume–Hu to generalized pairs. We also show that, for any lc generalized pair

such that

K_X+B+A+{\mathbf{M}}_X{\sim }_{\mathbb{R},Z}0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_X+B+A+\textbf{M}_X\sim _{{\mathbb {R}},Z}0$$\end{document}

and

B\ge 0,A\ge 0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B\ge 0,A\ge 0$$\end{document}

,

(X,B,\mathbf{M})/Z

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(X,B,\textbf{M})/Z$$\end{document}

has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon–Xu and Hashizume in the category of generalized pairs, and is later shown to be crucial to the proof of the existence of generalized lc flips in full generality.

We study connections among the ADM mass, positive harmonic functions, and capacity of the boundary on asymptotically flat 3-manifolds of nonnegative scalar curvature. We start with new formulae detecting the mass via positive harmonic functions. Then we derive a family of monotone quantities and geometric inequalities assuming the manifold has simple topology. As a first application, we observe several additional proofs of the 3-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions implying positivity of the mass via

-geometry of regions separating the boundary and

\infty

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\infty $$\end{document}

. A special case of such conditions shows if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by these findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio in the context of the Bartnik quasi-local mass.