Diameter Estimate in Geometric Flows
Abstract
:1. Introduction
- i.
- with . In this case, it follows thatand
- ii.
- where M and N are compact manifolds with the Riemannian metrics and h, respectively,are a family of smooth maps, is the intrinsic Laplacian of ψ, and is a smooth, positive and non-increasing function defined on . In this case, there holdsandSee Tadano [29] for a lower bound of the diameter for shrinking the Ricci-harmonic soliton.
- iii.
- Lorentzian mean curvature flow (see Holder [30] and Müller [17]). Let M be a compact space-like hyper-surface with in an ambient Lorentzian manifold L with , and letdenote a smooth immersion from M into L. We denote bya family of smooth immersions with andwhere and are the future-oriented, time-like normal vector and the mean curvature of the hyper-surface at the point , respectively. It follows thatwhere denotes the second fundamental form on . In this setup, one has
2. Proof of Main Theorem
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Fang, S.; Zheng, T. Diameter Estimate in Geometric Flows. Mathematics 2023, 11, 4659. https://doi.org/10.3390/math11224659
Fang S, Zheng T. Diameter Estimate in Geometric Flows. Mathematics. 2023; 11(22):4659. https://doi.org/10.3390/math11224659
Chicago/Turabian StyleFang, Shouwen, and Tao Zheng. 2023. "Diameter Estimate in Geometric Flows" Mathematics 11, no. 22: 4659. https://doi.org/10.3390/math11224659
APA StyleFang, S., & Zheng, T. (2023). Diameter Estimate in Geometric Flows. Mathematics, 11(22), 4659. https://doi.org/10.3390/math11224659