\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Existence of self-similar solutions of the inverse mean curvature flow

  • * Corresponding author: Kin Ming Hui

    * Corresponding author: Kin Ming Hui
Abstract Full Text(HTML) Related Papers Cited by
  • We will give a new proof of a recent result of P. Daskalopoulos, G. Huisken and J.R. King ([2] and reference [7] of [2]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $\mathbb{R}^n$, $n≥ 2$, of the form $u(x,t) = e^{λ t}f(e^{-λ t} x)$ for any constants $λ>\frac{1}{n-1}$ and $μ < 0$ such that $f(0) = μ$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $\text{div}\,\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}} \right) = \frac{1}{λ}·\frac{\sqrt{1+|\nabla f|^2}}{x·\nabla f-f}$ in $\mathbb{R}^n$, $f(0) = μ$, for any $λ>\frac{1}{n-1}$ and $μ < 0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $\lim_{r\to∞}\frac{rf_r(r)}{f(r)} = \frac{λ (n-1)}{λ (n-1)-1}$.

    Mathematics Subject Classification: Primary: 35K67, 35J75; Secondary: 53C44.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   B. Allen, Non-compact Solutions to Inverse Mean Curvature Flow in Hyperbolic Space, Ph.D. thesis, University of Tennessee, Knoxville, USA, 2016.
      P. Daskalopoulos and G. Huisken, Inverse mean curvature flow of entire graphs, arXiv: 1709.06665v1
      C. Gerhardt , Flow of nonconvex hypersurfaces into spheres, J. Differential Geom., 32 (1990) , 299-314. 
      G. Huisken  and  T. Ilmanen , The Riemannian Penrose inequality, Internat. Math. Res. Notices, 20 (1997) , 1045-1058.  doi: 10.1155/S1073792897000664.
      G. Huisken  and  T. Ilmanen , The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom., 59 (2001) , 353-437. 
      G. Huisken  and  T. Ilmanen , Higher regularity of the inverse mean curvature flow, J. Differential Geom., 80 (2008) , 433-451. 
      B. Lambert  and  J. Scheuer , The inverse mean curvature flow perpendicular to the sphere, Math. Ann., 364 (2016) , 1069-1093.  doi: 10.1007/s00208-015-1248-2.
      T. Marquardt , Inverse mean curvature flow for star-shaped hypersurfaces evolving in a cone, J. Geom. Anal., 23 (2013) , 1303-1313.  doi: 10.1007/s12220-011-9288-7.
      T. Marquardt , Weak solutions of the inverse mean curvature flow for hypersurfaces with boundary, J. Reine Angew. Math., 728 (2017) , 237-261.  doi: 10.1515/crelle-2014-0116.
      K. Smoczyk , Remarks on the inverse mean curvature flow, Asian J. Math., 4 (2000) , 331-335.  doi: 10.4310/AJM.2000.v4.n2.a3.
      J. Urbas , On the expansion of starshaped hypersurfaces by symmetric functions of their principle curvatures, Math. Z., 205 (1990) , 355-372.  doi: 10.1007/BF02571249.
  • 加载中
SHARE

Article Metrics

HTML views(792) PDF downloads(236) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return