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Register a new userQuantitative maximal volume entropy rigidity on Alexandrov spaces
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Lina Chen
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We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $epsilon(N, D)>0$, such that for $epsilon<epsilon(N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $geq -1$, $operatorname{diam}(X)leq D, h(X)geq N-1-epsilon$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity of cite{CRX} to Alexandrov spaces.
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In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karchers classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yaus gradient estimate for harmonic functions is also obtained on Alexandrov spaces.
Let $M$ be a compact $n$-manifold of $operatorname{Ric}_Mge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $rho>0$ such that for any $xin M$, the open $rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_rho,x^*)to (B_rho(x),x)$, has the maximal volume i.e., the volume of a $rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H e 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $tilde M$ is not collapsed for $H=1$ or $H e 1$ respectively. In the present paper, we will verify this conjecture for the case that Ricci curvature is also bounded above, while the above non-collapsing condition is not required.
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