Isoparametric foliations and the Pompeiu property

Luigi Provenzano; Alessandro Savo; Luigi Provenzano; Alessandro Savo

doi:10.3934/mine.2023031

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-27. doi: 10.3934/mine.2023031

Previous Article Next Article

Research article Special Issues

Isoparametric foliations and the Pompeiu property

Luigi Provenzano ,
Alessandro Savo ^,

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy

Received: 01 October 2021 Revised: 07 April 2022 Accepted: 08 April 2022 Published: 09 May 2022

A bounded domain $ \Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ \Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.
- Pompeiu property,
- isoparametric foliation,
- radial spectrum
Citation: Luigi Provenzano, Alessandro Savo. Isoparametric foliations and the Pompeiu property[J]. Mathematics in Engineering, 2023, 5(2): 1-27. doi: 10.3934/mine.2023031

Related Papers:

Abstract

A bounded domain $ \Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ \Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.

References

[1]	C. A. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Anal. Math., 37 (1980), 128–144. https://doi.org/10.1007/BF02797683 doi: 10.1007/BF02797683
[2]	C. A. Berenstein, L. Zalcman, Pompeiu's problem on spaces of constant curvature, J. Anal. Math., 30 (1976), 113–130. https://doi.org/10.1007/BF02786707 doi: 10.1007/BF02786707
[3]	C. A. Berenstein, L. Zalcman, Pompeiu's problem on symmetric spaces, Commentarii Mathematici Helvetici, 55 (1980), 593–621. https://doi.org/10.1007/BF02566709 doi: 10.1007/BF02566709
[4]	E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Annali di Matematica, 17 (1938), 177–191. https://doi.org/10.1007/BF02410700 doi: 10.1007/BF02410700
[5]	E. Cartan, Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z., 45 (1939), 335–367. https://doi.org/10.1007/BF01580289 doi: 10.1007/BF01580289
[6]	Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures, IV, J. Differential Geom., 115 (2020), 225–301. https://doi.org/10.4310/jdg/1589853626 doi: 10.4310/jdg/1589853626
[7]	R. Dalmasso, Le problème de Pompeiu, Séminaire de Théorie Spectrale et Géométrie, 17 (1998-1999), 69–79. https://doi.org/10.5802/tsg.206 doi: 10.5802/tsg.206
[8]	J. Ge, Z. Tang, Isoparametric functions and exotic spheres, J. Reine Angew. Math., 683 (2013), 161–180. https://doi.org/10.1515/crelle-2012-0005 doi: 10.1515/crelle-2012-0005
[9]	E. Giné M, The addition formula for the eigenfunctions of the Laplacian, Adv. Math., 18 (1975), 102–107. https://doi.org/10.1016/0001-8708(75)90003-1 doi: 10.1016/0001-8708(75)90003-1
[10]	S. Helgason, Differential geometry and symmetric spaces, New York-London: Academic Press, 1962.
[11]	T. Levi-Civita, Famiglie di superficie isoparametriche nell'ordinario spazio euclideo, Atti Accad. Naz. Lincei, Rend., VI. Ser., 26 (1937), 355–362.
[12]	H. F. Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann., 251 (1980), 57–71. https://doi.org/10.1007/BF01420281 doi: 10.1007/BF01420281
[13]	H. F. Münzner, Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel, Math. Ann., 256 (1981), 215–232. https://doi.org/10.1007/BF01450799 doi: 10.1007/BF01450799
[14]	K. Nomizu, Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc., 79 (1973), 1184–1188. https://doi.org/10.1090/S0002-9904-1973-13371-3 doi: 10.1090/S0002-9904-1973-13371-3
[15]	D. Pompeiu, Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables, C. R. Acad. Sci. Paris, 118 (1929), 1138–1139.
[16]	A. Savo, Heat flow, heat content and the isoparametric property, Math. Ann., 366 (2016), 1089–1136. https://doi.org/10.1007/s00208-015-1359-9 doi: 10.1007/s00208-015-1359-9
[17]	A. Savo, Geometric rigidity of constant heat flow, Calc. Var., 57 (2018), 156. https://doi.org/10.1007/s00526-018-1434-7 doi: 10.1007/s00526-018-1434-7
[18]	A. Savo, On the heat content functional and its critical domains, Calc. Var., 60 (2021), 167. https://doi.org/10.1007/s00526-021-02033-2 doi: 10.1007/s00526-021-02033-2
[19]	B. Segre, Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni, Atti Accad. Naz. Lincei, Rend., VI. Ser., 27 (1938), 203–207.
[20]	V. E. Shklover, Schiffer problem and isoparametric hypersurfaces, Rev. Mat. Iberoamericana, 16 (2000), 529–569. https://doi.org/10.4171/RMI/283 doi: 10.4171/RMI/283
[21]	T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385. https://doi.org/10.2969/jmsj/01840380 doi: 10.2969/jmsj/01840380
[22]	Z. Tang, Y. Xie, W. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, II, J. Funct. Anal., 266 (2014), 6174–6199. https://doi.org/10.1016/j.jfa.2014.02.024 doi: 10.1016/j.jfa.2014.02.024
[23]	Z. Tang, W. Yan, Isoparametric foliation and Yau conjecture on the first eigenvalue, J. Differential Geom., 94 (2013), 521–540. https://doi.org/10.4310/jdg/1370979337 doi: 10.4310/jdg/1370979337
[24]	L. Tchakaloff, Sur un problème de D. Pompéiu, (Bulgarian), Annuaire [Godišnik] Univ. Sofia. Fac. Phys.-Math. Livre 1., 40 (1944), 1–14.
[25]	G. Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, In: Handbook of differential geometry, Vol. I, Amsterdam: North-Holland, 2000,963–995. https://doi.org/10.1016/S1874-5741(00)80013-8
[26]	P. Ungar, Freak theorem about functions on a sphere, J. London Math. Soc., 29 (1954), 100–103. https://doi.org/10.1112/jlms/s1-29.1.100 doi: 10.1112/jlms/s1-29.1.100
[27]	H.-C. Wang, Two-point homogeneous spaces, Ann. Math., 55 (1952), 177–191. https://doi.org/10.2307/1969427 doi: 10.2307/1969427
[28]	Q. M. Wang, Isoparametric functions on Riemannian manifolds. I, Math. Ann., 277 (1987), 639–646. https://doi.org/10.1007/BF01457863 doi: 10.1007/BF01457863
[29]	S. A. Williams, A partial solution of the Pompeiu problem, Math. Ann., 223 (1976), 183–190. https://doi.org/10.1007/BF01360881 doi: 10.1007/BF01360881
[30]	S. A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J., 30 (1981), 357–369. https://doi.org/10.1512/iumj.1981.30.30028 doi: 10.1512/iumj.1981.30.30028
[31]	S. T. Yau, Problem section, In: Seminar on differential geometry, Princeton, N.J.: Princeton Univ. Press, 1982,669–706.
[32]	L. Zalcman, Analyticity and the Pompeiu problem, Arch. Rational Mech. Anal., 47 (1972), 237–254. https://doi.org/10.1007/BF00250628 doi: 10.1007/BF00250628
[33]	L. Zalcman, A bibliographic survey of the Pompeiu problem, In: Approximation by solutions of partial differential equations, Dordrecht: Springer, 1992,185–194. https://doi.org/10.1007/978-94-011-2436-2_17
[34]	A. Zettl, Sturm-Liouville theory, Providence, RI: American Mathematical Society, 2005. https://doi.org/10.1090/surv/121

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)