Article Contents
Article Contents

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

• * Corresponding author: Kin Ming Hui
• 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:

•  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.