On positive scalar curvature and moduli of curves

DOI10.4310/JDG/1549422104zbMATH Open1408.30043arXiv1506.03006OpenAlexW2964270118WikidataQ115165308 ScholiaQ115165308MaRDI QIDQ1717636

Yunhui Wu, Kefeng Liu

Publication date: 8 February 2019

Published in: Journal of Differential Geometry (Search for Journal in Brave)

Abstract: In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus

g

with

g g e q 2

does not admit any Riemannian metric

d s^{2}

of nonnegative scalar curvature such that

d s^{2} s u c c d s_{T}^{2}

where

d s_{T}^{2}

is the Teichm"uller metric. Our second result is the proof that any cover

M

of the moduli space

m a t h b b M_{g}

of a closed Riemann surface

S_{g}

does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm"uller metric, which implies a conjecture of Farb-Weinberger.

Full work available at URL: https://arxiv.org/abs/1506.03006

zbMATH Keywords

Riemannian metric scalar curvature Teichmüller metric moduli space of closed Riemann surfaces

Mathematics Subject Classification ID

Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) (32G15) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Teichmüller theory for Riemann surfaces (30F60)