Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

The isoperimetric inequality

Author(s): Robert Osserman
Journal: Bull. Amer. Math. Soc. 84 (1978), 1182-1238.
MSC (1970): Primary 52-02, 52A40; Secondary 26A84, 28A75, 35P15, 49F10, 53A10, 53C20
MathSciNet review: 0500557
Retrieve article in: PDF

References | Similar articles | Additional information

References:

L. Ahlfors, 1. Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157-194. MR 1555403

A. D. Aleksandrov (A. D. Alexandrow), 1. Isoperimetric inequalities for curved surfaces, Dokl. Akad. Nauk USSR 47 (1945), 235-238. MR 13543

A. D. Aleksandrov (A. D. Alexandrow), 2. Die innere Geometrie der konvexen Flächen, Akademie-Verlag, Berlin, 1955. MR 71041

A. D. Aleksandrov (A. D. Alexandrow), 3. Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad Univ. 13 (1958), 5-8; English transl., Amer. Math. Soc. Transl. (2) 21 (1962), pp. 412-416. MR 102114

A. D. Aleksandrov (A. D. Alexandrow), 4. A characteristic property of spheres, Annali Mat Pura Appl. 58 (1962), 303-315. MR 143162

A. D. Aleksandrov and V. V. Strel'tsov, 1. Estimates of the length of a curve on a surface, Dokl. Akad. Nauk USSR 93 (1953), 221-224. MR 60844

A. D. Aleksandrov and V. V. Strel'tsov, 2. The isoperimetric problem and estimates of the length of a curve on a surface, Proc. Steklov Inst. Math. 76 (1965), 67-80; English transl., Proc. Steklov Inst Math. (1967), 81-99. MR 202097

A. D. Aleksandrov and V. A. Zalgaller, 1. Two-dimensional manifolds of bounded curvature, Trudy Mat. Inst Steklov 63 (1962); English transl., Intrinsic geometry of surfaces, Amer. Math. Soc., Providence, R. I., 1967. MR 216434

A. D. Aleksandrov and V. A. Zalgaller, 2. Two-dimensional manifolds of bounded curvature, Trudy Mat Inst Steklov 76 (1965); English transl., Proc. Steklov Inst Math. (1967). MR 215209

H. Alexander, D. Hoffman and R. Osserman, 1. Area estimates for submanifolds of euclidean space, INDAM Symposia Math. Vol. 14 (1974), 445-455. MR 388253

W. K. Allard, 1. On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491. MR 307015

F. J. Almgren, Jr., 1. Three theorems on manifolds with bounded mean curvature, Bull. Amer. Math. Soc. 71 (1965), 755-756. MR 180952

F. J. Almgren, Jr., 2. The theory of varifolds, (mimeographed notes), Princeton, N. J., 1965.

T. Aubin, 1. Problèmes isopérimétriques et espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A 280 (1975), 279-281. MR 407905

T. Aubin, 2. Le problème de Yamabe concernant la courbure scalaire, C. R. Acad. Sci. Paris Sér. A 280 (1975), 721-724. MR 365407

T. Aubin, 3. Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), 573-598. MR 448404

T. Aubin, 4. Inégalités concernant la première valeur propre non nulle du laplacien pour certaines variétés riemanniennes, C. R. Acad. Sci. Paris Sér. A 281 (1975), 979-982. MR 413203

A. Baernstein, 1. Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139-169. MR 417406

A. Baernstein and B. A. Taylor, 1. Spherical rearrangements, subharmonic functions, and *-functions in n-space, Duke Math. J. 43 (1976), 245-268. MR 402083

T. F. Banchoff and W. F. Pohl, 1. A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971), 175-192. MR 305319

C. Bandle, 1. Konstruktion isoperimetrischer Ungleichungen der mathematischen Physik aus solchen der Geometrie, Comment. Math. Helv. 46 (1971), 182-213. MR 298551

C. Bandle, 2. Extremaleigenschaften von Kreissektoren und Halbkugeln, Comment. Math. Helv. 46 (1971), 356-380. MR 298552

C. Bandle, 3. A geometrical isoperimetric inequality and applications to problems of mathematical physics, Comment. Math. Helv 49 (1974), 496-511. MR 358100

C. Bandle, 4. On a differential inequality and its applications to geometry, Math. Z. 147 (1976), 253-261. MR 425773

Th. S. V. Bang, 1. An inequality for real functions of a real variable and its application to the prime number theorem, On Approximation Theory (Proc. Conf., Oberwolfach, August 1963), P. L. Butzer and J. Korevaar, editors, Birkhäuser, Basel, 1964, pp. 155-160. MR 182615

J. L. Barbosa and M. P. doCarmo, 1. On the size of a stable minimal surface inR3, Amer. J. Math. 98 (1976), 515-528. MR 413172

J. L. Barbosa and M. P. doCarmo, 2. Stable minimal surfaces inRn (to appear).

J. L. Barbosa and M. P. doCarmo, 3. A proof of a general isoperimetric inequality for surfaces (to appear).

E. F. Beckenbach and T. Radó, 1. Subharmonic functions and surfaces of negative curvature, Trans. Amer. Math. Soc. 35 (1933), 662-674. MR 1501708

D. C. Benson, 1. Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292-308. MR 202950

D. C. Benson, 2. Sharpened form of the isoperimetric inequality, Amer. Math. Monthly 77 (1970), 29-34. MR 257883

M. Berger, 1. Eigenvalues of the Laplacian, pp. 121-125 of Global Analysis, Proc. Sympos. Pure Math. 16, Amer. Math. Soc., Providence, R. I., 1970. MR 264549

M. Berger, 2. Sur les premières valeurs propres des variétés Riemanniennes, Compositio Math. 26 (1973), 129-149. MR 316913

M. Berger, 3. Geometry of the spectrum. I, pp. 129-152 of Differential Geometry, Proc. Sympos. Pure Math., vol. 27, part II, Amer. Math. Soc., Providence, R. I., 1975. MR 383459

M. Berger, P. Gauduchon and E. Mazet, 1. Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin, 1971. MR 282313

F. Bernstein, 1, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann. 60 (1905), 117-136. MR 1511289

A. S. Besicovitch, 1. On the definition and value of the area of a surface, Quart. J. Math., Oxford Ser. 16 (1945), 86-102. MR 14415

A. S. Besicovitch, 2. A variant of a classical isoperimetric problem, Quart J. Math. Oxford Ser. 20 (1949), 84-94. MR 30780

A. S. Besicovitch, 3. Parametric surfaces, Bull. Amer. Math. Soc. 56 (1950), 288-296. MR 36825

A. S. Besicovitch, 4. Variants of a classical isoperimetric problem, Quart. J. Math. Oxford Ser. 3 (1952), 42-49. MR 46667

W. Blaschke, 1. Kreis und Kugel, 2nd ed., Gruyter, Berlin, 1956. MR 77958

W. Blaschke, 2. Vorlesungen über Differentialgeometrie. I, 3rd ed., Springer-Verlag, Berlin, 1930.

W. Blaschke, 3. Vorlesungen über Integralgeometrie. I, Teubner, Leipzig, 1936; 3rd ed., VEB Deutscher Verlag, Berlin, 1955. MR 76373

W. Blaschke, 4. Griechische unde Anschauliche Geometrie, Oldenbourg, Munich, 1953. MR 57776

W. Blaschke and K. Leichtweiss, 1. Elementare Differentialgeometrie, Springer-Verlag, Berlin, 1973. MR 350630

W. Blaschke and H. Reichardt, 1. Einführung in die Differentialgeometrie, 2nd ed., Springer, Berlin, 1960. MR 116267

D. Bleecker and J. L. Weiner, 1. Extrinsic bounds onλ1, of Δ on a compact manifold, Comment. Math. Helv. 51 (1976), 601-609. MR 425839

G.Bol, 1. Isoperimetrische Ungleichung für Bereiche auf Flächen, Jber. Deutsch. Math.-Verein. 51 (1941), 219-257. MR 18858

E. Bombieri, 1. Theory of minimal surfaces, and a counter-example to the Bernstein conjecture in higher dimensions, Lecture notes, Courant Institute of Math., Spring, 1970.

E. Bombieri, E. deGiorgi, and M. Miranda, 1. Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal. 4 (1969), 255-267. MR 248647

E. Bombieri and E. Giusti, 1. Local estimates for the gradient of non-parametric surfaces of prescribed mean curvature, Comm. Pure Appl. Math. 26 (1973), 381-394. MR 344977

T. Bonnesen, 1. Über eine Verschärfung der isoperimetrischen Ungleichheit des Kreises in der Ebene und auf der Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper, Math. Ann. 84 (1921), 216-227. MR 1512031

T. Bonnesen, 2. Les problèmes des Isopérimètres et des Isépiphanes, Gauthier-Villars, Paris, 1929.

T. Bonnesen and W. Fenchel, 1. Theorie der konvexen Körper, Ergebnisse der Math, und ihrer Grenzgebiete, vol. 3, Springer-Verlag, Berlin, 1934; Berichtigter Reprint, 1974. MR 344997

S. Breuer and J. J. Roseman, 1. A lower bound for the eigenvalues of the elliptic Dirichlet problem for a general domain in terms of its characteristic dimension, SIAM J. Math. Anal. (to appear). MR 499821

Yu. D. Burago, 1. Isoperimetric inequalities in the theory of surfaces of bounded external curvature, Seminars in Math., V. A. Steklov Math. Inst., Leningrad, Vol. 10, 1968; English transl., Consultants Bureau, New York and London, 1970. MR 276905

Yu. D. Burago, 2. Note on the isoperimetric inequality on two-dimensional surfaces, Siberian Math. J. 14 (1973), 463-465.

Yu. D. Burago, 3. On the injectivity radius of surfaces with curvature bounded above, Ukrain Geometr. Sb. 21 (1978), 10-14. (Russian) MR 510649

Yu. D. Burago and V. A. Zalgaller, 1. Isoperimetric problems for regions on a surface having restricted width, Two-Dimensional Manifolds of Bounded Curvature, Trudy Mat. Inst. Steklov 76 (1965); 81-87; English transl., Proc. Steklov Inst. 76 (1965), 109-141 (1967). MR 202098

P. Buser, 1. Untersuchungen über den ersten Eigenwert des Laplace-Operators auf kompakten Flächen, Ph. D. Thesis, Basel, 1976.

P. Buser, 2. Riemannsche Flächen mit Eigenwerten in (0, ¼ ), Comment. Math. Helv. 52 (1977), 25-34. MR 434961

P. Buser, 3. Ueber eine Ungleichung von Cheeger, Math. Z. 158 (1978), 245-252. MR 478248

P. Buser, 4. Riemannsche Flaechen mit grosser Kragenweite, preprint, Bonn, 1977. MR 434961

P. Buser, 5. Ueber den ersten Eigenwert des Laplace-Operators auf kompakten Mannigfaltigkeiten nichtpositiver Krümmung, preprint, Bonn, 1977.

6.
P. Buser, Beispiele für λ, preprint, Bonn, 1977. MR 434961
T. Carleman, 1. Über ein Minimalproblem der mathematischen Physik, Math. Z. 1 (1918), 208-212. MR 1544292

T. Carleman, 2. Zur Theorie der Minimalflächen, Math. Z. 9 (1921), 154-160. MR 1544458

G. D. Chakerian, 1. The isoperimetric theorem for curves on minimal surfaces, Proc. Amer. Math. Soc. 69 (1978), 312-313. MR 474052

I. Chavel, 1. On A. Hurwitz' method in isoperimetric inequalities (to appear). MR 493885

I. Chavel and E. A. Feldman, 1. An optimal Poincaré inequality for convex domains of non-negative curvature, Arch. Rational Mech. Anal. 65 (1977), 263-273. MR 448457

I. Chavel and E. A. Feldman, 2. Isoperimetric inequalities on curved surfaces (to appear). MR 591721

J. Cheeger, 1. A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, A Symposium in Honor of Salomon Bochner, Robert C. Gunning, editor, Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199. MR 402831

S.-Y. Cheng, 1. Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), 289-297. MR 378001

S.-Y. Cheng, 2. Eigenfunctions and eigenvalues of Laplacian, Differential Geometry, Proc. Sympos. Pure Math., vol. 27, part II, Amer. Math. Soc., Providence, R. I., 1975, pp. 185-193. MR 378003

S.-Y. Cheng, 3. Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43-55. MR 397805

O. Chisini, 1. Sulla teoria elementare degli isoperimetri, Questioni riguardanti le mathematiche elementari, vol. 3., Bologna, 1927, pp. 201-310.

R. Courant, 1. Dirichlet's principle, conformal mapping and minimal surfaces, Interscience, New York, 1950. Springer-Verlag, New York, 1977 (reprint).

I. A. Danelich, 1. An estimate for the area of a surface of bounded absolute mean integral curvature in terms of its absolute mean integral curvature and the sum of the boundary curve lengths, Siberian Math. J. 7 (1966), 951-955. MR 209991

B. V. Dekster, 1. An inequality of the isoperimetric type for a domain in a Riemannian space, Mat Sb. 90 (1973); English transl., Math. USSR-Sb. 19 (1973), 257-274. MR 362159

V. I. Diskant, 1. Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference, Siberian Math. J. 13 (1972), 529-532.

V. I. Diskant, 2. Stability of the solution of the Minkowski equation, Siberian Math. J. 14 (1973), 466-469. MR 333988

V. I. Diskant, 3. Strengthening of an isoperimetric inequality, Siberian Math. J. 14 (1973), 608-611. MR 333989

V. I. Diskant, 4. A generalization of Bonnesen's inequalities, Dokl. Akad. Nauk SSSR 213 (1973) = Soviet Math. Dokl. 14 (1973), 1728-1731. MR 338925

M. P. doCarmo, 1. Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, N. J., 1976. MR 394451

M. Edelstein and B. Schwarz, 1. On the length of linked curves, Israel J. Math. 23 (1976), 94-95. MR 397558

C. Faber, 1. Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math.-Phys., Munich, 1923, 169-172.

H. Federer, 1. Geometric measure theory, Springer-Verlag, New York, 1969. MR 257325

H. Federer, 2. Colloquium Lectures, Amer. Math. Soc. Summer Meeting, Seattle, Wash., August 1977.

H. Federer and W. H. Fleming, 1. Normal and integral currents, Ann. of Math. 72 (1960), 458-520. MR 123260

J. Feinberg, 1. The isoperimetric inequality for doubly-connected minimal surfaces inR, Ph. D. Thesis, Stanford University, 1976.

J. Feinberg, 2. The isoperimetric inequality for doubly-connected minimal surfaces inR, J. d'Anal. Math. 32 (1977), 249-278. MR 461306

J. Feinberg, 3. Some Wirtinger-like inequalities (to appear). MR 547811

L. Fejes Tóth, 1. Lagerungen in der Ebene, auf der Kugel, und in Raum, Die Grundlehren der math. Wissenschaften, vol. 65, 2nd ed., Springer-Verlag, Berlin, 1972. MR 353117

F. Fiala, 1. Le problème des isopérimètres sur les surfaces ouvertes à courbure positive, Comment Math. Helv. 13 (1940/41), 293-346. MR 6422

R. Figiel, J. Lindenstrauss and V. D. Milman, 1. The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94. MR 445274

R. Finn, 1. On a class of conformal metrics, with application to differential geometry in the large, Comment. Math. Helv. 40 (1965), 1-30. MR 203618

Wm. J. Firey, 1. Affinities which preserve lower dimensional volumes, Amer Math. Monthly 72 (1965), 645. MR 1533306

W. H. Fleming, 1. On the oriented Plateau problem, Rend. Circ. Mat. Palermo 11 (1962), 69-90. MR 157263

L. E. Fraenkel and M. S. Berger, 1. A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. MR 422916

M. Gage, 1. Ph. D. thesis, Stanford University, 1978.

G. Galilei, 1. Two new sciences, translated, with Introduction and Notes, by Stillman Drake, Univ. of Wisconsin Press, Madison, Wisc, 1974. MR 497832

P. R. Garabedian, 1. Partial differential equations, Wiley, New York, 1964. MR 162045

T. Gasser and J. Hersch, 1. Über Eigenfrequenzen einer mehrfach zusammenhängenden Membran: Erweiterung von isoperimetrischen Sätzen von Pólya and Szegö, Z. Angew. Math. Phys. 19 (1968), 672-675. MR 234355

F. W. Gehring, 1. Symmetrization of rings in space. Trans. Amer. Math. Soc. 101 (1961), 499-519. MR 132841

F. W. Gehring, 2. Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 139735

F. W. Gehring, 3. Problem7.22, Symposium on Complex Analysis, Univ. of Kent (1973), Cambridge Univ. Press, Cambridge, 1974, p. 176. MR 387579

F. W. Gehring, 4. The Hausdorff measure of sets which link in euclidean space, Contributions to Analysis, A collection of papers dedicated to Lipman Bers, Academic Press, New York, 1974, pp. 159-167. MR 361008

H. Hadwiger, 1. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957. MR 102775

H. Hadwiger, 2. Gitterperiodische Punktmengen und Isoperimetrie, Monatsch. Math. 76 (1972), 410-418. MR 324550

K. Hanes, 1. Isoperimetric inequalities for manifolds with boundary, J. Differential Geometry 7 (1972), 525-534. MR 334081

G. H. Hardy, J. E. Littlewood, and G. Pólya, 1. Inequalities, Cambridge Univ. Press, Cambridge, 1952. MR 46395

P. Hartman, 1. Geodesic parallel coordinates in the large, Amer. Math. 86 (1964), 705-727. MR 173222

P. Hartman, 2. Some characterizations of the euclidean sphere, Nonlinear Analysis, Theory Methods and Applications 1 (1976), 37-48. MR 436046

W. K. Hayman, 1. Multivalent functions, Cambridge Univ. Press, Cambridge, 1958. MR 108586

W. K. Hayman, 2. Some bounds for principal frequency, Applicable Analysis (to appear). MR 492339

E. Heinz, 1. An inequality of isoperimetric type for surfaces of constant mean curvature, Arch. Rational Mech. Anal. 33 (1969), 155-168. MR 238188

E. Heinz, 2. Elementare Bemerkung zur isoperimetrischen Ungleichung inR3, Math. Z. 132 (1973), 319-322. MR 323976

E. Heinz and S. Hildebrandt, 1. The number of branch points of surfaces of bounded mean curvature, J. Differential Geometry 4 (1970), 227-235. MR 267495

J. Hersch, 1. Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A 270 (1970), 1645-1648. MR 292357

B. Herz and J. Kaapke, 1. Ein isoperimetrisches Problem mit Nebenbedingung, Elem. Math. 28 (1973), 63-65. MR 328778

S. Hildebrandt, 1. Über Flächen konstanter mittlerer Krümmung, Math. Z. 112 (1969), 107-144. MR 250206

S. Hildebrandt and H. Wente, 1. Variational problems with obstacles and a volume constraint, Math. Z. 135 (1973/74), 55-68. MR 365314

K. Hildén, 1. Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math. 18 (1976), 215-235. MR 409773

D. Hoffman, 1. Lower bounds on the first eigenvalue of the Laplacian on Riemannian manifolds (to appear). MR 574253

D. Hoffman, 2. Remarks on a geometric constant of Yau (to appear). MR 554517

D. Hoffman and J. Spruck, 1. Sobolev and isoperimetric inequalities for Riemannian submanifolds, Communications Pure Appl. Math. 27 (1974), 715-727; correction, ibid. 28 (1975), 765-766. MR 365424

C. O. Horgan and L. T. Wheeler, 1. Isoperimetric inequalities for the Dirichlet Eigenvalue problem, Quart. Appl. Math. 35 (1977), 406-409. MR 481624

H. Hopf, 1. Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4 (1950-51), 232-249. MR 40042

C. C. Hsiung, 1. Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220. MR 130637

A. Huber, 1. On the isoperimetric inequality on surfaces of variable Gaussian curvature, Ann. of Math. (2) 60 (1954), 237-247. MR 65946

A. Huber, 2. Zur isoperimetrischen Ungleichung auf gekrümmten Flächen, Acta. Math. 97 (1957), 95-101. MR 88750

A. Huber, 3. Zum potentialtheoretischen Aspekt der Alexandrowschen Flächentheorie, Comment. Math. Helv. 34 (1960), 99-126. MR 115147

A. Huber, 4. Zur Isoperimeterproblem auf vollstandigen Flächen mit summierbaren Gausscher Krümmung, Arch. Rational Mech. Anal. 24 (1967), 173-192. MR 222818

V. K. Ionin, 1. On isoperimetric and various other inequalities for a manifold of bounded curvature, Siberian Math. J. 10 (1969), 233-243. MR 240753

V. K. Ionin, 2. Isoperimetric inequalities for surfaces of negative curvature, Siberian Math. J. 13 (1972), 650-653. MR 309028

H. H. Johnson and J. Osaka, 1. An isoperimetric inequality for polyhedra, Amer. Math. Monthly 81 (1974), 58-61. MR 338935

M. Kac, 1. Can one hear the shape of a drum?Amer. Math. Monthly 73 (1966), 1-23. MR 201237

H. Karcher, 1. Umkreise und Inkreise konvexer Kurven in der sphärischen und der hyperbolischen Geometrie, Math. Ann. 177 (1968), 122-132. MR 226543

H. Karcher, 2. Anwendungen der Alexandrowschen Winkelvergleichssätze, Manuscripta Math. 2 (1970), 77-102. MR 263122

H. Kaul, 1. Isoperimetrische Ungleichung und Gauss-Bonnet-Formel für H-Flächen in Riemannschen Mannigfaltigkeiten, Arch. Rational Mech. Anal. 45 (1972), 194-221. MR 312403

H. Kaul, 2. Ein Einschliessungssatz für H-Flächen in Riemannschen Mannigfaltigkeiten, Manuscripta Math. 5 (1971), 103-112. MR 298594

H. Kaul, 3. Remarks on the isoperimetric inequality for multiply-connected H-surfaces, Math. Z. 128 (1972), 271-276. MR 312407

N. D. Kazarinoff, 1. Geometric inequalities, New Mathematical Library, Random House, New York, and Yale University, 1961. MR 130134

V. Klee, 1. Shapes of the future-some unsolved problems in geometry. Part II: Three dimensions (Viewer's Manual to accompany film), 1972.

M.-Th. Kohler-Jobin, 1. Démonstration de l'inégalité isopérimétrique $Płambda \sp{2}\geq \pi j\sp{4}\sb{0}/2$, conjecturée par Pólya et Szegö, C. R. Acad. Sci. Paris Sér. A 281 (1975), 119-121.

M.-Th. Kohler-Jobin, 2. Une inégalité isopérimétrique entre la fréquence fondamentale d'une membrane inhomogène et l'nergie d'équilibre du problème de Poisson correspondant, C. R. Acad. Sci. Paris Sér. A 283 (1976), 65-68. MR 421264

M.-Th. Kohler-Jobin, 3. Une propriété de monotonie isoperimétrique qui contient plusieurs théorèmes classiques, C. R. Acad. Sci. Paris Sér. A 284 (1977), 917-920. MR 434087

E. Krahn, 1. Über eine von Rayleigh formulierte Mininaleigenschaft des Kreises, Math. Ann. 94 (1925), 97-100. MR 1512244

E. Krahn, 2. Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1-44.

H. B. Lawson, Jr. and R. Osserman, 1. Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math. 139 (1977), 1-17. MR 452745

H. Lebesgue, 1. En marge du calcul des variations; Une introduction au calcul des variations et aux inégalités géométriques, Institut de Mathématiques, Geneva, 1963. MR 171195

H. Lewy, 1. Aspects of the Calculus of Variations (notes by J. W. Green), Univ. of Calif. Press, Berkeley, Calif., 1939.

H. Liebmann, 1. Ueber die Verbiegung der geschlossenen Flächen positiver Krümmung, Math. Ann. 53 (1900), 81-112. MR 1511083

T.-P. Lin, 1. Maximal area under constraint, Math. Mag. 50 (1977), 32-34. MR 430972

S. Lozinskii, 1. On subharmonic functions and their applications to the theory of surfaces, Izv. Akad. Nauk USSR Ser. Mat. 8 (1944), 175-194. MR 11371

J. Luttinger, 1. Generalized isoperimetric inequalities. I, II, III, J. Mathematical Phys. 14 (1973), 586-593, 1444-1447, 1448-1450. MR 337197

V. G. Maz'ya, 1. Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527-530; English transl., Soviet Math. Dokl. 1 (1960), 882-885. MR 126152

V. G. Maz'ya, 2. On certain integral inequalities for functions of many variables, Problemy Matematicheskogo Analiza, No. 3: Integral'nyei Differencial'nye Operatory. Differencial'nye Uravneniya, Leningrad 1972, pp. 33-68; English transl., J. Soviet Math. 1 (1973), 205-234.

H. P. McKean, 1. An upper bound to the spectrum ofΔ on a manifold of negative curvature, J. Differential Geom. 4 (1970), 359-366. MR 266100

P. McMullen, 1. Area preserving homeomorphisms, Elem. Math. 30 (1975), 86-87. MR 385720

J. H. Michael and L. M. Simon, 1. Sobolev and mean-value inequalities on generalized submanifolds of R, Comm. Pure Appl. Math. 26 (1973), 361-379. MR 344978

F. Minding, 1. Zur Theorie der Kurven kürzesten Umrings bei gegebenen Flächeninhalt auf krummen Flächen, J. Reine Angew. Math. 86 (1879), 279-289.

M. Miranda, 1. Disuguaglianze di Sobolev sulle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 69-79. MR 221350

D. S. Mitrinović, 1. Analytic inequalities, Springer, Berlin, 1970. MR 274686

J. Moser, 1. A sharp form of an inequality by N. Trudinger, Indiana Univ. J. Math. 20 (1970/71), 1077-1092. MR 301504

G. D. Mostow, 1. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 236383

Z. Nehari, 1. On the principal frequencies of a membrane, Pacific J. Math. 8 (1958), 285-293. MR 97606

J. C. C. Nitsche, 1. The isoperimetric inequality for multiply-connected minimal surfaces, Math. Ann. 160 (1965), 370-375. MR 185523

J. C. C. Nitsche, 2. An isoperimetric property of surfaces with moveable boundaries, Amer. Math. Monthly 77 (1970), 359-362. MR 257898

J. C. C. Nitsche, 3. Vorlesungen über Minimalflächen, Springer-Verlag, Berlin, 1975. MR 448224

R. Osserman, 1. Isoperimetric and related inequalities, Proc. Sympos. Pure Math. vol. 27, Amer. Math. Soc. Providence, R. I., 1975, pp. 207-215. MR 385686

R. Osserman, 2. Some remarks on the isoperimetric inequality and a problem of Gehring, J. Analyse Math. 30 (1976), 404-410. MR 445408

R. Osserman, 3. A note on Hayman's Theorem on the bass note of a drum, Comment. Math. Helv. 52 (1977), 545-555. MR 459099

R. Osserman, 4. Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly (to appear). MR 519520

R. Osserman and M. Schiffer, 1. Doubly-connected minimal surfaces, Arch. Rational Mech. Anal. 58 (1975), 285-307. MR 385687

T. Otsuki, 1. A remark on the Sobolev inequality for Riemannian manifolds, Proc. Japan Acad. 51 (1975), 785-789. MR 410814

L. E. Payne, 1. Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453-488. MR 218975

L. E. Payne and H. F. Weinberger, 1. Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl. 2 (1961), 210-216. MR 149735

J. Peetre, 1. A generalization of Courant's nodal line theorem, Math. Scand. 5 (1957), 15-20. MR 92917

H. Poincaré, 1. Sur un théorème de M. Liapounoff relatif à l'équilibre d'une masse fluide, C. R. Acad. Sci. Paris 104 (1887), 622-625.

H. Poincaré, 2. Figures d'équilibre d'une masse fluide (Leçons professées à la Sorbonne en 1900), edited by L. Dreyfus, Gauthier-Villars, Paris, 1902.

H. Poincaré, 3. Oeuvres, vol. 7.

G. Pólya, 1. Über geometrische Wahrscheinlichkeiten, Sitzungsber. Kaiserl. Akad. Wiss. Wien Math.-Natur. Kl., Abt. IIa 126 (1917), 1-10.

G. Pólya, 2. Induction and analogy in mathematics, Mathematics and Plausible Reasoning, vol. I., Princeton Univ. Press, Princeton, N. J., 1954. MR 66321

G. Pólya, 3. Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc. (N.S.) 24 (1960), 413-419. MR 133059

G. Pólya and G. Szegö, 1. Isoperimetric inequalities in mathematical physics, Ann. of Math. Studies, no. 27, Princeton Univ. Press, Princeton, N. J., 1951. MR 43486

G. Pólya and A. Weinstein, 1. On the torsional rigidity of multiply connected domains, Ann. of Math. (2) 52 (1950), 154-163. MR 40159

T. A. Porter, 1. A history of the classical isoperimetric problem, Contributions to the Calculus of Variations 1931-1932, Univ. of Chicago Press, Chicago, Illinois 1933.

P. Tolemy, 1. The Almagest

T. Radó, 1. A lemma on the topological index, Fund. Math. 27 (1936), 212-225.

T. Radó, 2. The isoperimetric inequality and the Lebesgue definition of surface area, Trans. Amer. Math. Soc. 41 (1947), 530-555. MR 21966

T. Radó, 3. Length and area, Amer. Math. Soc. Colloq. Publ, vol. 30, Amer. Math. Soc., Providence, R. I., 1948. MR 24511

(Lord) Rayleigh, 1. The theory of sound, MacMillan, New York, 1877, 1894; Dover, New York, 1945. MR 16009

W. T. Reid, 1. The isoperimetric inequality and associated boundary problems, J. Math. Mech. 8 (1959), 897-906. MR 130623

R. C. Reilly, 1. Applications of the integral of an invariant of the Hessian, Bull. Amer. Math. Soc. 82 (1976), 579-580. MR 415540

R. C. Reilly, 2. Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), 459-472. MR 474149

R. C. Reilly, 3. On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977), 525-533. MR 482597

R. C. Reilly, 4. Geometric applications of the solvability of Neumann problems on a Riemannian manifold (to appear). MR 592101

H. Sachs, 1. Ungleichungen für Umfang, Flächeninhalt und Trägheitsmoment konvexer Kurven, Acta. Math. Acad. Sci. Hungar. 11 (1960), 103-115. MR 140002

L. A. Santaló, 1. On the circle of maximum radius contained in a domain, Revista Un. Mat. Argentina 10 (1945), 155-162. (Spanish) MR 13550

L. A. Santaló, 2. Integral geometry on surfaces, Duke Math. J. 16 (1949), 361-375. MR 30228

L. A. Santaló, 3. Introduction to integral geometry, Actualitiés Sci. Indust. no. 1198, Hermann, Paris, 1953. MR 60840

L. A. Santaló, 4. Integral geometry and geometric probability, Addison-Wesley, Reading, Mass., 1977. MR 433364

E. Schmidt, 1. Über das isoperimetrische Problem im Raum von n Dimensionen, Math. Z 44 (1939), 689-788. MR 1545795

E. Schmidt, 2. Über die isoperimetrische Aufgabe im n-dimensionalen Raum konstanter negativer Krümmung. I. Die isoperimetrischen Ungleichungen in der hyperbolischen Ebene und für Rotationskörper im n-dimensionalen hyperbolischen Raum, Math. Z. 46 (1940), 204-230. MR 2196

E. Schmidt, 3. Die isoperimetrischen Ungleichungen auf der gewöhnlichen Kugel und für Rotationskörper im n-dimensionalen sphärischen Raum, Math. Z. 46 (1940), 743-794. MR 3733

E. Schmidt, 4. Über eine neue Methode zur Behandlung einer Klasse isoperimetrischer Aufgaben im Grossen, Math. Z. 47 (1942), 489-642. MR 16219

E. Schmidt, 5. Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl, Math. Z. 49 (1943/44), 1-109. MR 9127

R. Schoen, 1. Existence and regularity theorems for some geometrical variational problems, Ph. D. Thesis, Stanford Univ., 1977.

R. Schoen, S. Wolpert and S.-T. Yau, 1. On the first eigenvalue of a compact Riemann surface (to appear).

H. A. Schwarz, 1. Gesammelte Mathematische Abhandlungen, Springer-Verlag, Berlin, 1890.

S. Z. Šefel, 1. On the intrinsic geometry of saddle surfaces, Siberian Math. J. 5 (1964), 1382-1396. MR 175046

S. Z. Šefel, 2. Saddle surfaces which are bounded by a rectifiable curve, Soviet Math. Dokl. 6 (1965), 684-687.

J. Serrin, 1. A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318. MR 333220

M. Shiffman, 1. On the isoperimetric inequality for saddle surfaces with singularities, Studies and Essays presented to R. Courant, Interscience, New York, 1948, pp. 383-394. MR 23092

D. Singmaster and D. J. Soupporis, 1. A constrained isoperimetric problem, Math. Proc. Cambridge Philos. Soc. 83 (1978), 73-82. MR 470577

E. Sperner, 1. Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317-327. MR 340558

E. Sperner, 2. Symmetrisierung für Funktionen mehrerer reeler Variablen, Manuscripts Math. 11 (1974), 159-170. MR 328000

W. Spiegel, 1. Über die Symmetrisierung stetiger Funktionen im euklidischen Raum, Arch. Math. 24 (1973), 545-551. MR 412365

K. Steffen, 1. Isoperimetric inequalities and the problem of Plateau, Math. Ann. 222 (1976), 97-144. MR 417903

K. Steffen, 2. On the existence of surfaces with prescribed mean curvature and boundary, Math. Z. 146 (1976), 113-135. MR 394394

J. Steiner, 1. Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l'espace en général, J. Reine Angew. Math. 24 (1842), 93-152.

G. Stolzenberg, 1. Volumes, limits, and extensions of analytic varieties, Lecture notes in Math., vol. 19, Springer-Verlag, Berlin, 1966. MR 206337

G. Szegö., 1. Über einige Extremalaufgaben der Potentialtheorie, Math. Z. 31 (1930), 583-593. MR 1545137

G. Talenti, 1. Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372. MR 463908

V. A. Toponogov, 1. An isoperimetric inequality for surfaces whose Gaussian curvature is bounded above, Siberian Math. J. 10 (1969), 104-113. MR 238234

R. T. Waechter, 1. On hearing the shape of a drum; extensions to higher dimensions, Proc. Cambridge Philos. Soc. 72 (1972), 439-447. MR 304887

H. F. Weinberger, 1. Symmetrization in uniformly elliptic problems, pp. 424-428 of Studies in Mathematical Analysis and Related Topics; Essays in Honor of G. Pólya, Stanford Univ. Press, Stanford, Calif., 1962. MR 145191

H. F. Weinberger, 2. Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43 (1971), 319-320. MR 333221

J. L. Weiner, 1. A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974), 243-248. MR 380687

H. Wente, 1. An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344. MR 243467

H. Wente, 2. A general existence theorem for surfaces of constant mean curvature, Math. Z. 120 (1971), 277-288. MR 282300

H. Wente, 3. The differential equationΔx = 2H (x Λ x) with vanishing boundary values, Proc. Amer. Math. Soc. 50 (1975), 131-137. MR 374673

J. M. Wills, 1. Zum Verhältnis von Volumen zu Oberfläche bei konvexen Körpern, Arch. Math. 21 (1970), 557-560. MR 278192

S.-T. Yau, 1. Isoperimetric constants and the first eigenvalue of a compact manifold, Ann. Sci. École Norm. Sup. 8 (1975), 487-507. MR 397619


Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1970): 52-02, 52A40, 26A84, 28A75, 35P15, 49F10, 53A10, 53C20

Retrieve articles in all Journals with MSC (1970): 52-02, 52A40, 26A84, 28A75, 35P15, 49F10, 53A10, 53C20


Additional Information:

DOI: 10.1090/S0002-9904-1978-14553-4
PII: S 0002-9904(1978)14553-4




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia