Abstract
We say that a Riemannian manifold $(M, g)$ with a non-empty boundary $\partial M$ is a minimal orientable filling if, for every compact orientable $(\widetilde M,\tilde g)$ with $\partial \widetilde M=\partial M$, the inequality $ d_{\tilde g}(x,y) \ge d_g(x,y)$ for all $x,y\in\partial M$ implies $ \operatorname{vol}(\widetilde M,\tilde g) \ge \operatorname{vol}(M,g) .$ We show that if a metric $g$ on a region $M \subset \mathbf{R}^n$ with a connected boundary is sufficiently $C^2$-close to a Euclidean one, then it is a minimal filling. By studying the equality case $ \operatorname{vol}(\widetilde M,\tilde g) = \operatorname{vol}(M,g)$ we show that if $ d_{\tilde g}(x,y) = d_g(x,y)$ for all $x,y\in\partial M$ then $(M,g)$ is isometric to $(\widetilde M,\tilde g)$. This gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of Michel’s conjecture.
-
[AB] J. C. Alvarez-Paiva and G. Berck, "What is wrong with the Hausdorff measure in Finsler spaces," Adv. Math., vol. 204, pp. 647-663, 2006.
@article{AB,
author={Alvarez-Paiva, J. C. and Berck, G.},
TITLE={What is wrong with the Hausdorff measure in Finsler spaces},
JOURNAL={Adv. Math.},
VOLUME={204},
YEAR={2006},
PAGES={647--663},
MRNUMBER={2007g:53079},
ZBLNUMBER={1099.53018},
} -
[Bes]
A. S. Besicovitch, "On two problems of Loewner," J. London Math. Soc., vol. 27, pp. 141-144, 1952.
@article {Bes, MRKEY = {0047126},
AUTHOR = {Besicovitch, A. S.},
TITLE = {On two problems of {L}oewner},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {27},
YEAR = {1952},
PAGES = {141--144},
ISSN = {0024-6107},
MRCLASS = {27.2X},
MRNUMBER = {13,831d},
MRREVIEWER = {T. Rad{ó}},
DOI = {10.1112/jlms/s1-27.2.141},
ZBLNUMBER = {0046.05304},
} -
[Br]
L. E. J. Brouwer, "Beweis der invarianz des $n$-dimensionalen gebiets," Math. Ann., vol. 71, iss. 3, pp. 305-313, 1911.
@article {Br, MRKEY = {1511658},
AUTHOR = {Brouwer, L. E. J.},
TITLE = {Beweis der invarianz des {$n$}-dimensionalen gebiets},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {71},
YEAR = {1911},
NUMBER = {3},
PAGES = {305--313},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {Contributed Item},
MRNUMBER = {1511658},
DOI = {10.1007/BF01456846},
JFMNUMBER={42.0418.01},
} -
[BI95]
D. Burago and S. Ivanov, "On asymptotic volume of tori," Geom. Funct. Anal., vol. 5, iss. 5, pp. 800-808, 1995.
@article {BI95, MRKEY = {1354290},
AUTHOR = {Burago, D. and Ivanov, S.},
TITLE = {On asymptotic volume of tori},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {5},
YEAR = {1995},
NUMBER = {5},
PAGES = {800--808},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {53C20 (53C60)},
MRNUMBER = {96h:53041},
MRREVIEWER = {Joseph E. Borzellino},
DOI = {10.1007/BF01897051},
ZBLNUMBER = {0846.53043},
} -
[BI02]
D. Burago and S. Ivanov, "On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume," Ann. of Math., vol. 156, iss. 3, pp. 891-914, 2002.
@article {BI02, MRKEY = {1954238},
AUTHOR = {Burago, D. and Ivanov, S.},
TITLE = {On asymptotic volume of {F}insler tori, minimal surfaces in normed spaces, and symplectic filling volume},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {156},
YEAR = {2002},
NUMBER = {3},
PAGES = {891--914},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {53C60 (53C20)},
MRNUMBER = {2003k:53088},
MRREVIEWER = {Janko Latschev},
DOI = {10.2307/3597285},
ZBLNUMBER = {1040.53088},
} -
[BI04]
D. Burago and S. Ivanov, "Gaussian images of surfaces and ellipticity of surface area functionals," Geom. Funct. Anal., vol. 14, iss. 3, pp. 469-490, 2004.
@article {BI04, MRKEY = {2100668},
AUTHOR = {Burago, D. and Ivanov, S.},
TITLE = {Gaussian images of surfaces and ellipticity of surface area functionals},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {14},
YEAR = {2004},
NUMBER = {3},
PAGES = {469--490},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {53C60 (53C65)},
MRNUMBER = {2006b:53099},
MRREVIEWER = {Janko Latschev},
DOI = {10.1007/s00039-004-0465-8},
ZBLNUMBER = {1067.53042},
} -
[BCG]
G. Besson, G. Courtois, and S. Gallot, "Entropies et rigidités des espaces localement symétriques de courbure strictement négative," Geom. Funct. Anal., vol. 5, iss. 5, pp. 731-799, 1995.
@article {BCG, MRKEY = {1354289},
AUTHOR = {Besson, G. and Courtois, G. and Gallot, S.},
TITLE = {Entropies et rigidités des espaces localement symétriques de courbure strictement négative},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {5},
YEAR = {1995},
NUMBER = {5},
PAGES = {731--799},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {58F17 (53C21 53C35)},
MRNUMBER = {96i:58136},
MRREVIEWER = {Boris Hasselblatt},
DOI = {10.1007/BF01897050},
ZBLNUMBER = {0851.53032},
} -
[Croke91]
C. B. Croke, "Rigidity and the distance between boundary points," J. Differential Geom., vol. 33, iss. 2, pp. 445-464, 1991.
@article {Croke91, MRKEY = {1094465},
AUTHOR = {Croke, Christopher B.},
TITLE = {Rigidity and the distance between boundary points},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {33},
YEAR = {1991},
NUMBER = {2},
PAGES = {445--464},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C20},
MRNUMBER = {92a:53053},
MRREVIEWER = {Viktor Schroeder},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214446326},
ZBLNUMBER = {0729.53043},
} -
[Croke04] Geometric Methods in Inverse Problems and PDE Control, Croke, C. B., Lasiecka, I., Uhlmann, G., and Vogelius, M. S., Eds., New York: Springer-Verlag, 2004.
@book {Croke04, EDITOR={Croke, C. B. and Lasiecka, I. and Uhlmann, G. and Vogelius, M. S.},
MRKEY = {2164889},
TITLE = {Geometric Methods in Inverse Problems and {{\rm PDE}} Control},
SERIES = {The {\rm IMA} Volumes in Mathematics and its Applications},
NUMBER = {137},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {x+326},
ISBN = {0-387-40529-1},
MRCLASS = {00B25 (35-06 49-06 58-06)},
MRNUMBER = {2006d:00010},
ZBLNUMBER={1051.35003},
} -
[CDS]
C. B. Croke, N. S. Dairbekov, and V. A. Sharafutdinov, "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above," Trans. Amer. Math. Soc., vol. 352, iss. 9, pp. 3937-3956, 2000.
@article {CDS, MRKEY = {1694283},
AUTHOR = {Croke, Christopher B. and Dairbekov, Nurlan S. and Sharafutdinov, Vladimir A.},
TITLE = {Local boundary rigidity of a compact {R}iemannian manifold with curvature bounded above},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {352},
YEAR = {2000},
NUMBER = {9},
PAGES = {3937--3956},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {53C24 (53C22 58J53)},
MRNUMBER = {2000m:53054},
MRREVIEWER = {Dorothee Schueth},
DOI = {10.1090/S0002-9947-00-02532-0},
ZBLNUMBER = {0958.53027},
} -
[CK] C. B. Croke and B. Kleiner, "A rigidity theorem for simply connected manifolds without conjugate points," Ergodic Theory and Dynam. Systems, vol. 18, pp. 807-812, 1998.
@article{CK,
author = {Croke, Christopher B. and Kleiner, B.},
TITLE={A rigidity theorem for simply connected manifolds without conjugate points},
JOURNAL={Ergodic Theory and Dynam. Systems},
VOLUME={18},
YEAR={1998},
PAGES={807--812},
MRNUMBER={99j:53035},
MRREVIEWER={Raul Quiroga},
ZBLNUMBER={0924.53027},
} -
[Gromov]
M. Gromov, "Filling Riemannian manifolds," J. Differential Geom., vol. 18, iss. 1, pp. 1-147, 1983.
@article {Gromov, MRKEY = {697984},
AUTHOR = {Gromov, Mikhael},
TITLE = {Filling {R}iemannian manifolds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {18},
YEAR = {1983},
NUMBER = {1},
PAGES = {1--147},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C20 (53C21 57R99)},
MRNUMBER = {85h:53029},
MRREVIEWER = {Yu. Burago},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214509283},
ZBLNUMBER = {0515.53037},
} -
[HT]
R. D. Holmes and A. C. Thompson, "$n$-dimensional area and content in Minkowski spaces," Pacific J. Math., vol. 85, iss. 1, pp. 77-110, 1979.
@article {HT, MRKEY = {571628},
AUTHOR = {Holmes, R. D. and Thompson, A. C.},
TITLE = {{$n$}-dimensional area and content in {M}inkowski spaces},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {85},
YEAR = {1979},
NUMBER = {1},
PAGES = {77--110},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {52A40},
MRNUMBER = {81k:52023},
MRREVIEWER = {P. R. Goodey},
URL = {http://projecteuclid.org/getRecord?id=euclid.pjm/1102784083},
ZBLNUMBER = {0467.51007},
} -
[Ivanov] S. V. Ivanov, "On two-dimensional minimal fillings," Algebra i Analiz, vol. 13, iss. 1, pp. 26-38, 2001.
@article {Ivanov, MRKEY = {1819361},
AUTHOR = {Ivanov, S. V.},
TITLE = {On two-dimensional minimal fillings},
JOURNAL = {Algebra i Analiz},
FJOURNAL = {Rossiĭskaya Akademiya Nauk. Algebra i Analiz},
VOLUME = {13},
YEAR = {2001},
NUMBER = {1},
PAGES = {26--38},
ISSN = {0234-0852},
MRCLASS = {58E12 (53C20 53C60)},
MRNUMBER = {2002b:58016},
MRREVIEWER = {Nikolai K. Smolentsev},
ZBLNUMBER = {0995.58010},
} -
[Morgan]
F. Morgan, "Examples of unoriented area-minimizing surfaces," Trans. Amer. Math. Soc., vol. 283, iss. 1, pp. 225-237, 1984.
@article {Morgan, MRKEY = {735418},
AUTHOR = {Morgan, Frank},
TITLE = {Examples of unoriented area-minimizing surfaces},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {283},
YEAR = {1984},
NUMBER = {1},
PAGES = {225--237},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {49F22 (53C65)},
MRNUMBER = {85b:49079},
MRREVIEWER = {Jean E. Taylor},
DOI = {10.2307/1999999},
ZBLNUMBER = {0509.53007},
} -
[Michel]
R. Michel, "Sur la rigidité imposée par la longueur des géodésiques," Invent. Math., vol. 65, iss. 1, pp. 71-83, 1981.
@article {Michel, MRKEY = {636880},
AUTHOR = {Michel, R.},
TITLE = {Sur la rigidité imposée par la longueur des géodésiques},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {65},
YEAR = {1981},
NUMBER = {1},
PAGES = {71--83},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58D99 (53C22)},
MRNUMBER = {83d:58021},
MRREVIEWER = {Francesco Mercuri},
DOI = {10.1007/BF01389295},
ZBLNUMBER = {0471.53030},
} -
[PU]
L. Pestov and G. Uhlmann, "Two dimensional compact simple Riemannian manifolds are boundary distance rigid," Ann. of Math., vol. 161, iss. 2, pp. 1093-1110, 2005.
@article {PU, MRKEY = {2153407},
AUTHOR = {Pestov, Leonid and Uhlmann, Gunther},
TITLE = {Two dimensional compact simple {R}iemannian manifolds are boundary distance rigid},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {161},
YEAR = {2005},
NUMBER = {2},
PAGES = {1093--1110},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {53C24 (53C22)},
MRNUMBER = {2006c:53038},
MRREVIEWER = {Rodney Josu{é} Biezuner},
DOI = {10.4007/annals.2005.161.1093},
ZBLNUMBER = {1076.53044},
} -
[Santalo] L. A. Santaló, Integral Geometry and Geometric Probability, Reading, MA: Addison-Wesley Publ. Co., 1976.
@book {Santalo, MRKEY = {0433364},
AUTHOR = {Santal{ó},
Luis A.},
TITLE = {Integral Geometry and Geometric Probability},
PUBLISHER = {Addison-Wesley Publ. Co.},
ADDRESS={Reading, MA},
YEAR = {1976},
PAGES = {xvii+404},
MRCLASS = {53C65 (60D05)},
MRNUMBER = {55 \#6340},
MRREVIEWER = {W. J. Firey},
ZBLNUMBER = {0342.53049},
} -
[SU]
P. Stefanov and G. Uhlmann, "Boundary rigidity and stability for generic simple metrics," J. Amer. Math. Soc., vol. 18, iss. 4, pp. 975-1003, 2005.
@article {SU, MRKEY = {2163868},
AUTHOR = {Stefanov, Plamen and Uhlmann, Gunther},
TITLE = {Boundary rigidity and stability for generic simple metrics},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {18},
YEAR = {2005},
NUMBER = {4},
PAGES = {975--1003},
ISSN = {0894-0347},
MRCLASS = {53C24 (58J32)},
MRNUMBER = {2006h:53031},
MRREVIEWER = {Rodney Josu{é} Biezuner},
DOI = {10.1090/S0894-0347-05-00494-7},
ZBLNUMBER = {1079.53061},
} -
[Th] A. C. Thompson, Minkowski Geometry, Cambridge: Cambridge Univ. Press, 1996.
@book {Th, MRKEY = {1406315},
AUTHOR = {Thompson, A. C.},
TITLE = {Minkowski Geometry},
SERIES = {Encycl. Math. Appl.},
NUMBER = {63},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1996},
PAGES = {xvi+346},
ISBN = {0-521-40472-X},
MRCLASS = {52-02 (51-02)},
MRNUMBER = {97f:52001},
MRREVIEWER = {W. J. Firey},
ZBLNUMBER = {0868.52001},
}