2013, 8(1): 1-8. doi: 10.3934/nhm.2013.8.1

Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow

1. 

Department of Mathematics, UW Madison, Madison, WI53705, United States

Received  March 2012 Revised  October 2012 Published  April 2013

We provide formal matched asymptotic expansions for ancient convex solutions to MCF. The formal analysis leading to the solutions is analogous to that for the generic MCF neck pinch in [1].
    For any $p, q$ with $p+q=n$, $p\geq1$, $q\geq2$ we find a formal ancient solution which is a small perturbation of an ellipsoid. For $t\to-\infty$ the solution becomes increasingly astigmatic: $q$ of its major axes have length $\approx\sqrt{2(q-1)(-t)}$, while the other $p$ axes have length $\approx \sqrt{-2t\log(-t)}$.
    We conjecture that an analysis similar to that in [2] will lead to a rigorous construction of ancient solutions to MCF with the asymptotics described in this paper.
Citation: Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
References:
[1]

S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow,, J. Reine Angew. Math., 482 (1997), 15.

[2]

Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities,, J. Reine Angew. Math., 672 (2012).

[3]

Sigurd Angenent, Shrinking doughnuts,, Nonlinear diffusion equations and their equilibrium states, 7 (1992), 21.

[4]

Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow,, J. Differential Geom., 84 (2010), 455.

[5]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

show all references

References:
[1]

S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow,, J. Reine Angew. Math., 482 (1997), 15.

[2]

Sigurd Angenent, Cristina Caputo and Dan Knopf, Minimally invasive surgery for Ricci flow singularities,, J. Reine Angew. Math., 672 (2012).

[3]

Sigurd Angenent, Shrinking doughnuts,, Nonlinear diffusion equations and their equilibrium states, 7 (1992), 21.

[4]

Panagiota Daskalopoulos, Richard Hamilton and Natasa Sesum, Classification of compact ancient solutions to the curve shortening flow,, J. Differential Geom., 84 (2010), 455.

[5]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Differential Geom., 23 (1986), 69.

[1]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441

[2]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

[3]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963

[4]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[5]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[6]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169

[7]

Georgi I. Kamberov. Recovering the shape of a surface from the mean curvature. Conference Publications, 1998, 1998 (Special) : 353-359. doi: 10.3934/proc.1998.1998.353

[8]

G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11.

[9]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[10]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[11]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041

[12]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[13]

Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010

[14]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[15]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[16]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

[17]

Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155

[18]

Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409

[19]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159

[20]

Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]