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Register a new userEigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound
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We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove t
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Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one.
We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincar{e}-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is equivalent to Ricci curvature bounded below by $K$. On the other hand, it was observed in cite{Ri} that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy $MCP(0,5)$. In this paper, we give sufficient conditions for a $2n+1$ dimensional weakly Sasakian manifold to satisfy $MCP(0,2n+3)$. This extends the above mentioned result on the Heisenberg group in cite{Ri}.
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.
On complete noncompact Riemannian manifolds with non-negative Ricci curvature, Li-Schoen proved the uniform Poincare inequality for any ge odesic ball. In this note, we obtain the sharp lower bound of the first Dirichlet eigenvalue of such geodesic balls, which implies the sharp Poincare inequality for geodesic balls.
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