Combinatorial p-th R-curvature Ricci Flows and Calabi Flows on Surfaces

Chunlei Liu

Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 523 -545.

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Frontiers of Mathematics ›› 2024, Vol. 20 ›› Issue (3) : 523 -545. DOI: 10.1007/s11464-023-0094-x
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Combinatorial p-th R-curvature Ricci Flows and Calabi Flows on Surfaces

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Abstract

In this paper, we study the convergence of the solutions to combinatorial p-th R-curvature Ricci flows and combinatorial p-th R-curvature Calabi flows on surfaces. R-curvature was introduced by Ge [Int. Math. Res. Not. IMRN, 2017, 2017(11): 3510–3527] which is a modification of the well-known discrete Gaussian curvature on triangulated manifolds. We show that the long time convergence of the solutions to combinatorial p-th R-curvature Ricci flows on surfaces is equivalent to the existence of constant R-curvature metrics. Furthermore, we show that the solutions to combinatorial p-th R-curvature Calabi flows on surfaces in the Euclidean background geometry and hyperbolic background geometry have the long time convergence if and only if there exist constant R-curvature metrics.

Keywords

Combinatorial p-th R-curvature Ricci flow / combinatorial p-th R-curvature Calabi flow / convergence / constant R-curvature metric / potential functional

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Chunlei Liu. Combinatorial p-th R-curvature Ricci Flows and Calabi Flows on Surfaces. Frontiers of Mathematics, 2024, 20(3): 523-545 DOI:10.1007/s11464-023-0094-x

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