| Abstract: |
| In this talk, we introduce the evolution problem of area-preserving and length-preserving flows with the positive power and negative power of anisotropic curvature. For the case of the positive power anisotropic curvature flows, by utilizing Tso$^{\prime}$s method, we establish time-independent upper bound estimate for curvature and then derive estimates for the derivatives of curvature. Both flows are shown to exist globally and converge to the boundary of the homothety of Wulff shapes. For the case of negative power, we apply Tso$^{\prime}$s method to establish the time-independent lower bound estimate of curvature for the area-preserving curvature flow. And, we apply Bernstein$^{\prime}$s estimate or Moser iteration method to establish the lower bound of curvature for the length-preserving curvature flow. Under the assumption of global existence, it is shown that the flows converge to the boundary of the homothety of Wulff shapes. |
|