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doi: 10.3934/dcdss.2020153

Remarks on mean curvature flow solitons in warped products

1. 

Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini, 50, Milano, 20133, Italy

2. 

Scuola Normale Superiore, Piazza dei Cavalieri, 7, Pisa, 56124, Italy

3. 

Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini, 50, Milano, 20133, Italy

* Corresponding author

Dedicated to Patrizia Pucci on her 65th birthday

Received  May 2018 Revised  December 2018 Published  November 2019

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

Citation: Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020153
References:
[1]

L. J. Alías, J. H. de Lira and M. Rigoli, Mean curvature flow solitons in the presence of conformal vector fields, preprint, arXiv: 1707.07132. Google Scholar

[2]

L. J. Alías, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-24337-5.  Google Scholar

[3]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.  Google Scholar

[4]

C. Bao and Y. G. Shi, Gauss maps of translating solitons of mean curvature flow, Proc. Amer. Math. Soc., 142 (2014), 4333-4339.  doi: 10.1090/S0002-9939-2014-12209-X.  Google Scholar

[5]

J. Barta, Sur la vibration fundamentale d'une membrane, C. R. Acad. Sci., 204 (1937), 472-473.   Google Scholar

[6]

G. Pacelli BessaL. F. Pessoa and M. Rigoli, Vanishing theorems, higher order mean curvatures and index estimates for self-shrinkers, Israel J. Math., 226 (2018), 703-736.  doi: 10.1007/s11856-018-1703-3.  Google Scholar

[7]

B. Bianchini, L. Mari, P. Pucci and M. Rigoli, On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds, preprint, arXiv: 1801.02102. Google Scholar

[8]

B. BianchiniL. Mari and M. Rigoli, Spectral radius, index estimates for Schrödinger operators and geometric applications, J. Funct. Anal., 256 (2009), 1769-1820.  doi: 10.1016/j.jfa.2009.01.021.  Google Scholar

[9]

B. BianchiniL. Mari and M. Rigoli, Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds, J. Funct. Anal., 268 (2015), 1-72.  doi: 10.1016/j.jfa.2014.10.016.  Google Scholar

[10]

B. Bianchini, L. Mari and M. Rigoli, On some aspects of oscillation theory and geometry, Mem. Amer. Math. Soc., 225 (2013). doi: 10.1090/s0065-9266-2012-00681-2.  Google Scholar

[11]

J.-P. Bourguignon, The "magic" of Weitzenböck formulas, Variational methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 4 (1990), 251–271.  Google Scholar

[12]

R. Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z., 178 (1981), 501-508.  doi: 10.1007/BF01174771.  Google Scholar

[13]

H.-D. CaoY. Shen and S. H. Zhu, The structure of stable minimal hypersurfaces in Rn+1, Math. Res. Lett., 4 (1997), 637-644.  doi: 10.4310/MRL.1997.v4.n5.a2.  Google Scholar

[14]

X. Cheng and D. T. Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc., 141 (2013), 687-696.  doi: 10.1090/S0002-9939-2012-11922-7.  Google Scholar

[15]

X. ChengT. Mejia and D. T. Zhou, Simons-type equation for f-minimal hypersurfaces and applications, J. Geom. Anal., 25 (2015), 2667-2686.  doi: 10.1007/s12220-014-9530-1.  Google Scholar

[16]

J. ClutterbuckO. C. Schnürer and F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations, 29 (2007), 281-293.  doi: 10.1007/s00526-006-0033-1.  Google Scholar

[17]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow Ⅰ: Generic singularities, Ann. of Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.  Google Scholar

[18]

B. Devyver, On the finiteness of the Morse index for Schrödinger operators, Manuscripta Math., 139 (2012), 249-271.  doi: 10.1007/s00229-011-0522-1.  Google Scholar

[19]

Q. DingY. L. Xin and L. Yang, The rigidity theorems of self shrinkers via Gauss maps, Adv. Math., 303 (2016), 151-174.  doi: 10.1016/j.aim.2016.08.019.  Google Scholar

[20]

M. P. do CarmoH. B. Lawson and Jr ., On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J., 50 (1983), 995-1003.  doi: 10.1215/S0012-7094-83-05041-X.  Google Scholar

[21]

A. FarinaL. Mari and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations, 38 (2013), 1818-1862.  doi: 10.1080/03605302.2013.795969.  Google Scholar

[22]

D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., 82 (1985), 121-132.  doi: 10.1007/BF01394782.  Google Scholar

[23]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.  Google Scholar

[24]

S. Fornari and J. Ripoll, Killing fields, mean curvature, translation maps, Illinois J. Math, 48 (2004), 1385-1403.  doi: 10.1215/ijm/1258138517.  Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[26]

T. Hasanis and D. Koutroufiotis, A property of complete minimal surfaces, Trans. Amer. Math. Soc., 281 (1984), 833-843.  doi: 10.1090/S0002-9947-1984-0722778-5.  Google Scholar

[27]

Y. Higuchi, A remark on exponential growth and the spectrum of the Laplacian, Kodai Math. J., 24 (2001), 42-47.  doi: 10.2996/kmj/1106157294.  Google Scholar

[28]

D. A. HoffmanR. Osserman and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in R3 and R4, Comment. Math. Helv., 57 (1982), 519-531.  doi: 10.1007/BF02565874.  Google Scholar

[29]

D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727.  doi: 10.1002/cpa.3160270601.  Google Scholar

[30]

D. Impera and M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, Geom. Dedicata, 178 (2015), 21-47.  doi: 10.1007/s10711-014-9999-6.  Google Scholar

[31]

D. Impera and M. Rimoldi, Rigidity results and topology at infinity of translating solitons of the mean curvature flow, Commun. Contemp. Math., 19 (2017), 1750002, 21 pp. doi: 10.1142/S021919971750002X.  Google Scholar

[32]

M. Kanai, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math., 6 (1983), 143-151.  doi: 10.3836/tjm/1270214332.  Google Scholar

[33]

P. W.-K. Li, Harmonic Functions and Applications to Complete Manifolds, XIV Escola de Geometria Diferencial, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2006.  Google Scholar

[34]

P. Li and J. P. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett., 9 (2002), 95-103.  doi: 10.4310/MRL.2002.v9.n1.a7.  Google Scholar

[35]

L. MariP. Mastrolia and M. Rigoli, A note on Killing fields and CMC hypersurfaces, J. Math. Anal. Appl., 431 (2015), 919-934.  doi: 10.1016/j.jmaa.2015.06.016.  Google Scholar

[36]

W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math., 75 (1978), 219-226.  doi: 10.2140/pjm.1978.75.219.  Google Scholar

[37]

M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14 (1962), 333-340.  doi: 10.2969/jmsj/01430333.  Google Scholar

[38] B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.   Google Scholar
[39]

J. Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations, 15 (1974), 541-550.  doi: 10.1016/0022-0396(74)90072-2.  Google Scholar

[40]

S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Progress in Mathematics, 266. Birkhäuser Verlag, Basel, 2008.  Google Scholar

[41]

M. Rigoli and A. G. Setti, Liouville type theorems for ϕ-subharmonic functions, Rev. Mat. Iberoamericana, 17 (2001), 471-520.  doi: 10.4171/RMI/302.  Google Scholar

[42]

A. Rocha, Essential spectrum of the weighted Laplacian on noncompact manifolds and applications, Geom. Dedicata, 186 (2017), 197-219.  doi: 10.1007/s10711-016-0186-9.  Google Scholar

[43]

K. Smoczyk, A relation between mean curvature flow solitons and minimal submanifolds, Math. Nachr, 229 (2001), 175-186.  doi: 10.1002/1522-2616(200109)229:1<175::AID-MANA175>3.0.CO;2-H.  Google Scholar

[44]

Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117 (1965), 251-275.  doi: 10.1090/S0002-9947-1965-0174022-6.  Google Scholar

[45]

X.-J. Wang, Convex solutions to the mean curvature flow, Ann. of Math. (2), 173 (2011), 1185–1239. doi: 10.4007/annals.2011.173.3.1.  Google Scholar

show all references

References:
[1]

L. J. Alías, J. H. de Lira and M. Rigoli, Mean curvature flow solitons in the presence of conformal vector fields, preprint, arXiv: 1707.07132. Google Scholar

[2]

L. J. Alías, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-24337-5.  Google Scholar

[3]

S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations, 2 (1994), 101-111.  doi: 10.1007/BF01234317.  Google Scholar

[4]

C. Bao and Y. G. Shi, Gauss maps of translating solitons of mean curvature flow, Proc. Amer. Math. Soc., 142 (2014), 4333-4339.  doi: 10.1090/S0002-9939-2014-12209-X.  Google Scholar

[5]

J. Barta, Sur la vibration fundamentale d'une membrane, C. R. Acad. Sci., 204 (1937), 472-473.   Google Scholar

[6]

G. Pacelli BessaL. F. Pessoa and M. Rigoli, Vanishing theorems, higher order mean curvatures and index estimates for self-shrinkers, Israel J. Math., 226 (2018), 703-736.  doi: 10.1007/s11856-018-1703-3.  Google Scholar

[7]

B. Bianchini, L. Mari, P. Pucci and M. Rigoli, On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds, preprint, arXiv: 1801.02102. Google Scholar

[8]

B. BianchiniL. Mari and M. Rigoli, Spectral radius, index estimates for Schrödinger operators and geometric applications, J. Funct. Anal., 256 (2009), 1769-1820.  doi: 10.1016/j.jfa.2009.01.021.  Google Scholar

[9]

B. BianchiniL. Mari and M. Rigoli, Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds, J. Funct. Anal., 268 (2015), 1-72.  doi: 10.1016/j.jfa.2014.10.016.  Google Scholar

[10]

B. Bianchini, L. Mari and M. Rigoli, On some aspects of oscillation theory and geometry, Mem. Amer. Math. Soc., 225 (2013). doi: 10.1090/s0065-9266-2012-00681-2.  Google Scholar

[11]

J.-P. Bourguignon, The "magic" of Weitzenböck formulas, Variational methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 4 (1990), 251–271.  Google Scholar

[12]

R. Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z., 178 (1981), 501-508.  doi: 10.1007/BF01174771.  Google Scholar

[13]

H.-D. CaoY. Shen and S. H. Zhu, The structure of stable minimal hypersurfaces in Rn+1, Math. Res. Lett., 4 (1997), 637-644.  doi: 10.4310/MRL.1997.v4.n5.a2.  Google Scholar

[14]

X. Cheng and D. T. Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc., 141 (2013), 687-696.  doi: 10.1090/S0002-9939-2012-11922-7.  Google Scholar

[15]

X. ChengT. Mejia and D. T. Zhou, Simons-type equation for f-minimal hypersurfaces and applications, J. Geom. Anal., 25 (2015), 2667-2686.  doi: 10.1007/s12220-014-9530-1.  Google Scholar

[16]

J. ClutterbuckO. C. Schnürer and F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial Differential Equations, 29 (2007), 281-293.  doi: 10.1007/s00526-006-0033-1.  Google Scholar

[17]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow Ⅰ: Generic singularities, Ann. of Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.  Google Scholar

[18]

B. Devyver, On the finiteness of the Morse index for Schrödinger operators, Manuscripta Math., 139 (2012), 249-271.  doi: 10.1007/s00229-011-0522-1.  Google Scholar

[19]

Q. DingY. L. Xin and L. Yang, The rigidity theorems of self shrinkers via Gauss maps, Adv. Math., 303 (2016), 151-174.  doi: 10.1016/j.aim.2016.08.019.  Google Scholar

[20]

M. P. do CarmoH. B. Lawson and Jr ., On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J., 50 (1983), 995-1003.  doi: 10.1215/S0012-7094-83-05041-X.  Google Scholar

[21]

A. FarinaL. Mari and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations, 38 (2013), 1818-1862.  doi: 10.1080/03605302.2013.795969.  Google Scholar

[22]

D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., 82 (1985), 121-132.  doi: 10.1007/BF01394782.  Google Scholar

[23]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.  Google Scholar

[24]

S. Fornari and J. Ripoll, Killing fields, mean curvature, translation maps, Illinois J. Math, 48 (2004), 1385-1403.  doi: 10.1215/ijm/1258138517.  Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[26]

T. Hasanis and D. Koutroufiotis, A property of complete minimal surfaces, Trans. Amer. Math. Soc., 281 (1984), 833-843.  doi: 10.1090/S0002-9947-1984-0722778-5.  Google Scholar

[27]

Y. Higuchi, A remark on exponential growth and the spectrum of the Laplacian, Kodai Math. J., 24 (2001), 42-47.  doi: 10.2996/kmj/1106157294.  Google Scholar

[28]

D. A. HoffmanR. Osserman and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in R3 and R4, Comment. Math. Helv., 57 (1982), 519-531.  doi: 10.1007/BF02565874.  Google Scholar

[29]

D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727.  doi: 10.1002/cpa.3160270601.  Google Scholar

[30]

D. Impera and M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, Geom. Dedicata, 178 (2015), 21-47.  doi: 10.1007/s10711-014-9999-6.  Google Scholar

[31]

D. Impera and M. Rimoldi, Rigidity results and topology at infinity of translating solitons of the mean curvature flow, Commun. Contemp. Math., 19 (2017), 1750002, 21 pp. doi: 10.1142/S021919971750002X.  Google Scholar

[32]

M. Kanai, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math., 6 (1983), 143-151.  doi: 10.3836/tjm/1270214332.  Google Scholar

[33]

P. W.-K. Li, Harmonic Functions and Applications to Complete Manifolds, XIV Escola de Geometria Diferencial, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2006.  Google Scholar

[34]

P. Li and J. P. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett., 9 (2002), 95-103.  doi: 10.4310/MRL.2002.v9.n1.a7.  Google Scholar

[35]

L. MariP. Mastrolia and M. Rigoli, A note on Killing fields and CMC hypersurfaces, J. Math. Anal. Appl., 431 (2015), 919-934.  doi: 10.1016/j.jmaa.2015.06.016.  Google Scholar

[36]

W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math., 75 (1978), 219-226.  doi: 10.2140/pjm.1978.75.219.  Google Scholar

[37]

M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14 (1962), 333-340.  doi: 10.2969/jmsj/01430333.  Google Scholar

[38] B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.   Google Scholar
[39]

J. Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations, 15 (1974), 541-550.  doi: 10.1016/0022-0396(74)90072-2.  Google Scholar

[40]

S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Progress in Mathematics, 266. Birkhäuser Verlag, Basel, 2008.  Google Scholar

[41]

M. Rigoli and A. G. Setti, Liouville type theorems for ϕ-subharmonic functions, Rev. Mat. Iberoamericana, 17 (2001), 471-520.  doi: 10.4171/RMI/302.  Google Scholar

[42]

A. Rocha, Essential spectrum of the weighted Laplacian on noncompact manifolds and applications, Geom. Dedicata, 186 (2017), 197-219.  doi: 10.1007/s10711-016-0186-9.  Google Scholar

[43]

K. Smoczyk, A relation between mean curvature flow solitons and minimal submanifolds, Math. Nachr, 229 (2001), 175-186.  doi: 10.1002/1522-2616(200109)229:1<175::AID-MANA175>3.0.CO;2-H.  Google Scholar

[44]

Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117 (1965), 251-275.  doi: 10.1090/S0002-9947-1965-0174022-6.  Google Scholar

[45]

X.-J. Wang, Convex solutions to the mean curvature flow, Ann. of Math. (2), 173 (2011), 1185–1239. doi: 10.4007/annals.2011.173.3.1.  Google Scholar

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