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• W2807869135 abstract "We introduce combinatorial $alpha$-curvature for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical combinatorial curvature on polyhedral surfaces. Then we prove the global rigidity of $alpha$-curvature with respect to the discrete conformal factors. To study the corresponding Yamabe problem of $alpha$-curvature, we introduce the combinatorial $alpha$-Yamabe flow and combinatorial $alpha$-Calabi flow for piecewise linear metrics on surfaces. To handle the possible singularities along the flows, we do surgery on the flows by flipping. Then we prove that if $alphachi(S)leq 0$, there exists a piecewise linear metric with constant combinatorial $alpha$-curvature on a polyhedral surface $S$, which is a parameterized generalization of Gu-Luo-Sun-Wu's discrete uniformization theorem. We further prove that the combinatorial $alpha$-Yamabe flow and $alpha$-Calabi flow with surgery exists for all time and converges to a piecewise linear metric with constant combinatorial $alpha$-curvature for any initial piecewise linear metric if $alphachi(S)leq 0$." @default.
• W2807869135 created "2018-06-21" @default.
• W2807869135 creator A5014888334 @default.
• W2807869135 date "2018-06-11" @default.
• W2807869135 modified "2023-09-22" @default.
• W2807869135 title "Combinatorial $alpha$-curvatures and $alpha$-flows on polyhedral surfaces, I" @default.
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• W2807869135 hasPublicationYear "2018" @default.
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