Width estimate and doubly warped product

Zhu, Jintian

doi:10.1090/tran/8263

Width estimate and doubly warped product
HTML articles powered by AMS MathViewer

by Jintian Zhu PDF

Trans. Amer. Math. Soc. 374 (2021), 1497-1511 Request permission

Abstract:

In this paper, we give an affirmative answer to Gromov’s conjecture (Geom. Funct. Anal. 28 (2018), pp. 645–726, Conjecture E) by establishing an optimal Lipschitz lower bound for a class of smooth functions on connected orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that if the optimal bound is attained the given manifold must be a quotient space of $\mathbf R^2\times (-c,c)$ with some doubly warped product metric. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in $3$-spheres with positive sectional curvatures.

References

Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830. MR 2765731, DOI 10.4310/CAG.2010.v18.n4.a6
Otis Chodosh, Michael Eichmair, and Vlad Moraru, A splitting theorem for scalar curvature, Comm. Pure Appl. Math. 72 (2019), no. 6, 1231–1242. MR 3948556, DOI 10.1002/cpa.21803
Misha Gromov, Metric inequalities with scalar curvature, Geom. Funct. Anal. 28 (2018), no. 3, 645–726. MR 3816521, DOI 10.1007/s00039-018-0453-z
Misha Gromov, Four lectures on scalar curvature, 2019, preprint.
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
Fernando C. Marques and André Neves, Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J. 161 (2012), no. 14, 2725–2752. MR 2993139, DOI 10.1215/00127094-1813410
Vladimir S. Matveev, Can we make a Finsler metric complete by a trivial projective change?, Recent trends in Lorentzian geometry, Springer Proc. Math. Stat., vol. 26, Springer, New York, 2013, pp. 231–242. MR 3064804, DOI 10.1007/978-1-4614-4897-6_{1}0
Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. MR 2260928
Xin Zhou and Jonathan Zhu, Existence of hypersurfaces with prescribed mean curvature I—generic min-max, Camb. J. Math. 8 (2020), no. 2, 311–362. MR 4091027, DOI 10.4310/CJM.2020.v8.n2.a2
Jintian Zhu, Rigidity of area-minimizing $2$-spheres in $n$-manifolds with positive scalar curvature, Proc. Amer. Math. Soc. 148 (2020), no. 8, 3479–3489. MR 4108854, DOI 10.1090/proc/15033