The mathematics of F. J. Almgren, Jr.
 Brian White
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Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a moreorless chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.
 Almgren, F. J. (1962) The homotopy groups of the integral cycle groups. Topology 1: pp. 257299 CrossRef
 Almgren, F. J. (1964) An isoperimetric inequality. Proc. Am. Math. Soc 15: pp. 284285 CrossRef
 Almgren, F. J. (1965) Three theorems on manifolds with bounded mean curvature. Bull. Am. Math. Soc. 71: pp. 755756 CrossRef
 Almgren, F. J. (1965) Mass continuous cochains are differential forms. Proc. Am. Math. Soc. 16: pp. 12911294 CrossRef
 F. J. Almgren, JrThe Theory of Varifolds. A variational calculus in the large for the kdimensional are integrand, Multilithed notes, Princeton University Library, 178, 1965.
 F. J. Almgren, JrPlateau’s Problem. An Invitation to Varifold Geometry. Benjamin, W.A., Ed. New York, 1966.
 Almgren, F. J. (1966) Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84: pp. 277292 CrossRef
 Almgren, F. J. (1967) Existence and regularity of solutions to elliptic calculus of variations problems among surfaces of varying topological type and singularity structure. Bull. Am. Math. Soc 73: pp. 576680 CrossRef
 Almgren, F. J. (1968) Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. 87: pp. 321391 CrossRef
 Almgren, F. J. (1969) Measure theoretic geometry and elliptic variational problems. Bull. Am. Math. Soc 75: pp. 285304 CrossRef
 Almgren, F. J. (1969) A maximum principle for elliptic variational problems. J. Functional Anal. 4: pp. 380389 CrossRef
 Almgren, F. J. (1974) Measure theoretic geometry and elliptic variational problems. Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis, (Tbilisi, 1971), (in Russian) II: pp. 307324
 Almgren, F. J. (1971) Geometric measure theory and elliptic variational problems. Actes du Congrès International des Mathématiciens, (Nice, 1970). GauthierVillars, Paris, pp. 813819
 Almgren, F. J., Allard, W.K. (1973) An introduction to regularity theory for parametric elliptic variational problems. Partial differential equations. Proc. Symp. Pure Math. XXIII: pp. 231260
 Almgren, F. J. (1974) Geometric variational problems from a measuretheoretic point of view. Global analysis and its applications. Internat. Atomic Energy Agency, Viena, pp. 122
 Almgren, F. J. (1973) Geometric measure theory and elliptic variational problems. Geometric Measure Theory and Minimal Surfaces. Ediziono Cremonese, Rome, pp. 31117
 Almgren, F. J. (1974) The structure of limit varifolds associated with minimizing sequences of mappings. Symposia Mathematica. Academic Press, London
 Almgren, F. J. (1975) Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Bull. Am. Math. Soc. 81: pp. 151154 CrossRef
 Almgren, F. J. (1976) Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4: pp. viiiviii
 Almgren, F. J., Allard, W.K. (1976) The structure of stationary onedimensional varifolds with positive density. Inv. Math. 34: pp. 8397 CrossRef
 F. J. Almgren, Jr with Taylor, J.E. The geometry of soap films and soap bubbles,Sci. Am., 82–93, July (1976).
 Almgren, F. J., Thurston, W.P. (1977) Examples of unknotted curves which bound only surfaces’ of high genus with their convex hulls. Ann. Math. 105: pp. 527538 CrossRef
 Almgren, F. J., Schoen, R., Simon, L. (1977) Regularity and singularity estimates of hypersurfaces minimizing parametric elliptic variational integrals. Acta Math. 139: pp. 217265 CrossRef
 Almgren, F. J., Simon, L. (1979) Existence of embedded solutions of Plateau’s problem. Annali Scuola Normale Superiore de Pisa (Series IV) VI: pp. 447495
 Almgren, F. J. Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal submanifolds and geodesics. In: Obata, M. eds. (1978) Proceedings of the JapanU.S. Seminar on Minimal Submanifolds including Geodesies. Kaigai Publishings, Tokyo, Japan, pp. 16
 F. J. Almgren, Jr with Taylor, J.E. Descriptive geometry in the calculus of variations, Proceedings of the International Congress on Descriptive Geometry (Vancouver, 1978),Engineering Design Graphics J., (1979).
 Almgren, F. J. Minimal surfaces: tangent cones, singularities, and topological types. In: Lehto, O. eds. (1980) Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Acad. Sci. Fennica, Helsinki, pp. 767770
 Almgren, F. J., Allard, W.K. (1981) On the radial behavior of minimal surfaces and the uniqueness of their tangent cones. Ann. Math. 113: pp. 215265 CrossRef
 Almgren, F. J., Thurston, R.N. (1981) Liquid crystals and geodesies. J. Phys. 42: pp. 413417
 Almgren, F. J. (1982) Minimal Surfaces. McGrawHill, New York
 Almgren, F. J. (1982) Minimal surface forms. The Mathematical Intelligencer 4: pp. 164172
 Almgren, F. J. (1983) Approximation of rectifiable currents by Lipschitz Qvalued functions, Seminar on Minimal Submanifolds. Ann. Math. Studies. Princeton University Press, Princeton, NJ, pp. 243259
 Almgren, F. J. (1983) Qvalued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Am. Math. Soc. 8: pp. 327328 CrossRef
 F. J. Almgren, Jr Qvalued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, (V. Scheffer and J. Taylor, Eds.),World Scientific, to appear. Currently available electronically at http://www.math.princeton.edu/~scheffer.
 Almgren, F. J., Super, B. (1984) Multiple valued functions in the geometric calculus of variations. Astérisque 118: pp. 1332
 Almgren, F. J. (1985) Optimal isoperimetric inequalities. Bull. Am. Math. Soc 13: pp. 123126 CrossRef
 Almgren, F. J. (1986) Optimal isoperimetric inequalities. Indiana V. Math. J. 35: pp. 451547 CrossRef
 Almgren, F. J. (1986) Deformations and multiple valued functions, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984). Proc. Sympos. Pure Math., Am. Math. Soc., Providence, RI 44: pp. 29130
 Almgren, F. J. (1987) Applications of multiple valued functions. Geometric Modeling: Algorithms and New Trends. SIAM, Philadelphia, pp. 4354
 F. J. Almgren, Jr Spherical symmetrization,Proceedings of the International Workshop on Integral Functions in the Calculus of Variations (Trieste, 1985), Rend. Circ. Mat. Palermo (2) Suppl., 11–25, 1987.
 Almgren, F. J., Taylor, J.E. Optimal crystal shapes. In: Concus, P., Finn, R. eds. (1987) Variational Methods for Free Surface Interfaces. SpringerVerlag, New York, pp. 111
 Almgren, F. J., Lieb, E.H. (1987) Singularities of energy minimizing maps from the ball to the sphere. Bull. Am. Math. Soc. 17: pp. 304306 CrossRef
 Almgren, F. J., Browder, W., Lieb, E.H. Coarea, liquid crystals, and minimal surfaces. In: Chern, S.S. eds. (1988) Partial Differential Equations. SpringerVerlag, New York, pp. 122 CrossRef
 Almgren, F. J., Lieb, E.H. (1988) Singularities of energy minimizing maps from the ball to the sphere. Ann. Math. 128: pp. 483530 CrossRef
 Almgren, F. J., Lieb, E.H. (1989) Counting singularities in liquid crystals. Symposia Mathematica. Academic Press, London, pp. 103118
 Almgren, F. J., Lieb, E.H. (1989) Symmetric decreasing rearrangement can be discontinuous. Bull. Am. Math. Soc. 20: pp. 177180 CrossRef
 Almgren, F. J., Lieb, E.H. (1989) Symmetric rearrangement is sometimes continuous. J. Am. Math. Soc. 2: pp. 683773 CrossRef
 Almgren, F. J., Gurtin, M. (1989) A mathematical contribution to Gibbs’s analyses of fluid phases in equilibrium. Partial Differential Equations and the Calculus of Variations, Progr. Nonlinear Differential Equations Appl.. Birkäuser, Boston, pp. 928
 Almgren, F. J., Browder, W. Homotopy with holes and minimal surfaces. In: Lawson, B., Tenenblat, K. eds. (1991) Differential Geometry. Longman Scientific & Technical, Harlow, pp. 1523
 F. J. Almgren, Jr What can geometric measure theory do for several complex variables? Proceedings of the Several Complex Variables Year at the MittagLeffler Institute (Stockholm, 1987–1988), Princeton University Press Math. Notes (38), Princeton, NJ, 8–21, 1993.
 Almgren, F. J. (1993) The geometric calculus of variations and modelling natural phenomena, Statistical thermodynamics and differential geometry of microstructured materials (Minneapolis, MN, 1991). IMA Vol. Math. Appl.. SpringerVerlag, New York, pp. 15
 Almgren, F. J. (1991) Multifunctions modv, Geometric analysis and computer graphics (Berkeley, CA, 1988). Math. Sci. Res. Inst. Publ.. SpringerVerlag, New York, pp. 117
 Almgren, F. J., Lieb, E.H. (1992) The (non)continuity of symmetric decreasing rearrangement, Proceedings of the conference on geometry of solutions to PDE (Cortona, 1988). Symposia Mathematica. Academic Press, Boston, MA, pp. XXXXXX
 F. J. Almgren, Jr Computing soap films and crystals,Computing Optimal Geometries, video report,Am. Math. Soc., 1991.
 Almgren, F. J., Sullivan, J. (1992) Visualization of soap bubble geometries. Leonardo 25: pp. 267271 CrossRef
 F. J. Almgren, Jr with Taylor, J.E. and Wang, L. A variational approach to motion by weighted mean curvature, Computational Crystal Growers Workshop Selected Lectures in Mathematics,Am. Math. Soc., 9–12, (1992).
 Almgren, F. J., Taylor, J.E., Wang, L. (1993) Curvature driven flows: A variational approach. S1AM J. Control and Optimization 31: pp. 387438 CrossRef
 Almgren, F. J. (1992) Questions and answers about area minimizing surfaces and geometric measure theory, Differential Geometry: partial differential equations on manifolds, (Los Angeles, 1990). Proc. Symposia Pure Math., Am. Math. Soc 51: pp. 2953
 Almgren, F. J., Taylor, J.E. (1995) Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differential Geom. 42: pp. 122
 Almgren, F. J., Taylor, J.E. (1995) Optimal geometry in equilibrium and growth, Symposium in Honor of Benoit Mandelbrot (Curaçao, 1995). Fractals 3: pp. 713723 CrossRef
 Almgren, F. J. (1993) Questions and answers about geometric evolution processes and crystal growth. The Gelfand Mathematical Seminars. Gelfand Math. Sem., Birkhäuser, Boston, pp. 19
 F. J. Almgren, Jr with Wang, L. Mathematical existence of crystal growth with GibbsThomson curvature effects,J. Geom. Anal, (to appear).
 F. J. Almgren, Jr with Rivin, I. The mean curvature integral is invariant under bending, 1–21, Geometry and topology monographs #1, University of Warwick published electronically: www.maths.warwick.ac.uk/gt/main/ml
 Almgren, F. J., Taylor, J. (1996) Soap bubble clusters: the Kelvin problem. Forma 11: pp. 199207
 F. J. Almgren, JrGlobal Analysis. preprint, (survey/expository).
 F. J. Almgren, Jr Isoperimetric inequalities for anisotropic surface energies, unfinished manuscript.
 F. J. Almgren, Jr A new look at flat chains modn, unfinished manuscript.
 Adams, D., Simon, L. (1988) Rates of asymptotic convergence near isolated singularities of geometric extrema. Indiana U. Math. J. 37: pp. 225254 CrossRef
 Allard, W.K. (1972) On the first variation of a varifold. Annal. Math. 95: pp. 417491 CrossRef
 Allard, W.K. (1986) An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled. Am. Math. Soc. Proc. Symp. Pure Math., (Arcata) 44: pp. 128
 Ambrosio, L., Braides, A. (1990) Functionals defined on partitions in sets of finite perimeter I, II. J. Math. Pures Appl. 69: pp. 285333
 Anzellotti, G. (1983) On the c1,α regularity of ωminima of quadratic functionals. Boll. Un. Math. Ital. C (6) 2: pp. 195212
 Bombieri, E., Giorgi, E., Giusti, E. (1969) Minimal cones and the Bernstein problem. Invent. Math. 7: pp. 243268 CrossRef
 Bombieri, E. (1982) Regularity theory for almost minimal currents. Arch. Rational Mech. Anal 78: pp. 99130 CrossRef
 Brakke, K. (1977) The Motion of a Surface by its Mean Curvature.. Princeton University Press, Princeton, NJ
 Brakke, K. (1992) The Surface Evolver. Experimental Mathematics 1: pp. 141165
 Brakke, K. (1995) Soap films and covering spaces. J. Geom. Anal 5: pp. 445514 CrossRef
 Calabi, E. (1967) Minimal immmersions of surfaces in Euclidean spheres. J. Diff. Geometry 1: pp. 111125
 Cao, J. and Escobar, J.F. An isoperimetric comparison theorem for PLmanifolds of nonpositive curvature, preprint, (revised version), 1998.
 Chang, S. (1988) Twodimensional area minimizing integral currents are classical minimal surfaces. J. Am. Math. Soc 1: pp. 699778 CrossRef
 Choe, J. (1990) The isoperimetric inequality for a minimal surface with radially connected boundary. Ann. ScuolaNorm. Sup. Pisa Cl. Sci. 17: pp. 583593
 Choe, J. (1996) Three sharp isoperimetric inequalities for stationary varifolds and area minimizing flat chains modk. Kodai Math. J. 19: pp. 177190 CrossRef
 Croke, C. (1984) A sharp fourdimensional isoperimetric inequality. Comment. Math. Helv. 59: pp. 187192 CrossRef
 Pauw, T. (1998) On SBV dual. Indiana U. Math. J. 47: pp. 99121 CrossRef
 Evans, L.C. Quasiconvexity and partial regularity in the calculus of variations,Arch. Rational Mech. Anal,95, 227–252.
 Federer, H. (1986) Flat chains with positive densities. Indiana U. Math. J. 35: pp. 413424 CrossRef
 Federer, H., Fleming, W. (1960) Normal and integral currents. Ann. Math. 72: pp. 458520 CrossRef
 Garafolo, N., Lin, F.H. (1986) Monotonicity properties of variational integrals,A p weights unique continuation. Indiana U. Math. J. 35: pp. 245267 CrossRef
 Giaquinta, M., Giusti, E. (1984) Quasiminima. Ann. Inst. H. Poincaré Anal. Non Linéare 1: pp. 79107
 Giaquinta, M., Modica, G., Souček, J. (1989) Cartesian currents, weak diffeomorphisms, and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal 106: pp. 97159 CrossRef
 Giaquinta, M., Modica, G., and Souček, J. Cartesian currents in the calculus of variations, 2 volumes, SpringerVerlag, 1998.
 Gromov, M., Schoen, R. (1992) Harmonic maps into singular spaces and padic superrigidity for lattices in groups of rank one. Inst. Hautes Etudes Sci. Publ. Math. 76: pp. 165246 CrossRef
 Gulliver, R., White, B. (1989) The rate of convergence of a harmonic map at a singular point. Math. Ann. 283: pp. 539549 CrossRef
 Hamilton, R. (1982) Threemanifolds with positive ricci curvature. J. Diff. Geom. 17: pp. 255306
 Hardt, R. (1977) On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand. Communications in P.D.E. 2: pp. 11631232 CrossRef
 Hardt, R., Lin, F.H. (1993) Harmonic maps into round cones and singularities of nematic liquid crystals. Math. Z. 213: pp. 575593 CrossRef
 Hardt, R., Kinderlehrer, D., Lin, F.H. (1988) Stable defects of minimizers of constrained variational principles. Ann. Inst. H. Poincaré, Anal. Nonl. 5: pp. 297322
 Hubbard, J.H. (1980) On the convex hull genus of space curves. Topology 19: pp. 203208 CrossRef
 Hutchinson, J. (1986) C1,α multiple function regularity and tangent cone behaviour for varifolds with mean curvature inL p. Am. Math. Soc. Proc. Symp. Pure Math., (Arcata) 44: pp. 281306
 Kleiner, B. (1992) An isoperimetric comparison theorem. Invent. Math. 108: pp. 3747 CrossRef
 Li, P., Schoen, R., Yau, S.T. (1984) On the isoperimetric inequality for minimal surfaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11: pp. 237244
 Lin, F.H. (1991) Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 44: pp. 287308 CrossRef
 Luckhaus, S. (1990) Solutions for the twophase Stefan problem with GibbsThomson law for the melting temperature. Euro. J. App. Math. 1: pp. 101111
 Mackenzie, D. (1997) Fred Almgren (1933–1997): lover of mathematics, family, and life’s adventures. Notices Am. Math. Soc 44: pp. 11021106
 Micallef, M., White, B. (1995) The structure of branch points in area minimizing surfaces and in pseudoholomorphic curves. Ann. Math. 141: pp. 3585 CrossRef
 Michael, J., Simon, L. (1973) Sobolev and meanvalue inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math. 26: pp. 361379 CrossRef
 Morgan, F. (1984) Area minimizing currents bounded by higher multiples of curves. Rend. Circ. Mat. Palermo 33: pp. 3746 CrossRef
 Morgan, F. (1991) The cone over the Clifford torus in R4 is Φminimizing. Math. Ann. 289: pp. 341354 CrossRef
 Morrey, C.B. (1968) Nonlinear elliptic systems. J. Math. Meck 17: pp. 649670
 Osserman, R. (1978) The isoperimetric inequality. Bull. Am. Math. Soc 84: pp. 11821238 CrossRef
 Pitts, J. (1977) Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Princeton University Press, Princeton, NJ
 Schoen, R., Simon, L. (1982) A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana U. Math. J. 31: pp. 415434 CrossRef
 Simons, J. (1968) Minimal varieties in Riemannian manifolds. Ann. Math. 88: pp. 62105 CrossRef
 Simon, L. (1983) Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. 118: pp. 525571 CrossRef
 Simon, L. Isolated singularities of extrema of geometric variational problems, in Harmonic mappings and minimal immersions (Montecatini, 1984),Springer Lecture Notes in Math.,1161, 206–277.
 Simon, L. (1993) Cylindrical tangent cones and the singular set of minimal submanifolds. J. Diff. Geom. 38: pp. 585652
 Simon, L. (1995) Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps. Surveys in Differential Geometry. Internat. Press, Cambridge, MA, pp. 246305
 Simon, L. (1995) Rectifiability of the singular set of energy minimizing maps. Calc. Var. Part. Diff. Eq. 3: pp. 165
 Simon, L. (1996) Theorems on regularity and singularity of energy minimizing maps, Based on lecture notes by Norbert Hungerbühler. Lectures in Mathematics ETH Zürich. Birkhäuser, SpringerVerlag, Basel
 Solomon, B. (1984) A new proof of the closure theorem for integral currents. Indiana U. Math. J. 33: pp. 393418 CrossRef
 Solomon, B. (1990) The harmonic analysis of cubic isoparametric hypersurfaces I, II. Am. J. Math. 112: pp. 157241 CrossRef
 Taylor, J.E. (1973) Regularity of the singular sets of twodimensional areaminimizing flat chains modulo 3 in R3. Invent. Math. 22: pp. 119159 CrossRef
 Taylor, J.E. (1976) The structure of singularities in soapbubblelike and soapfilmlike minimal surfaces. Ann. Math. 103: pp. 489539 CrossRef
 Taylor, J.E. (1998) Frederick Justin Almgren, 1933–1997. Journal of Geometric Analysis 8: pp. 673674
 Weil, A. (1926) Sur les surfaces a courbure negative. C. R. Acad. Sci. 182: pp. 10691071
 White, B. (1983) Tangent cones to 2dimensional areaminimizing integral currents are unique. Duke Math. J. 50: pp. 143160 CrossRef
 White, B. (1984) The least area bounded by multiples of a curve. Proc. Am. Math. Soc. 90: pp. 230232 CrossRef
 White, B. (1986) A regularity theorem for minimizing hypersurfaces modp, Geometric Measure Theory and the Calculus of Variations. Am. Math. Soc., Proc. Symp. Pure Math. 44: pp. 413427
 White, B. (1989) A new proof of the compactness theorem for integral currents. Comm. Math. Helv. 64: pp. 207220 CrossRef
 White, B. (1989) Some Recent Developments in Differential Geometry. Math. Intelligencer 11: pp. 4147
 White, B. (1992) Nonunique tangent maps at isolated singularities of harmonic maps. Bull. Am. Math. Soc 26: pp. 125129 CrossRef
 White, B. (1996) Existence of leastenergy configurations of immiscible fluids. J. Geom. Anal. 6: pp. 151161
 White, B. (1997) Statification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Ang. Math. 488: pp. 135
 White, B. (1997) Classical area minimizing surfaces with real analytic boundaries. Acta Math. 179: pp. 295305 CrossRef
 White, B. The deformation theorem for flat chains,Acta Math., (to appear).
 White, B. Rectifiability of flat chains,Annals of Math., (to appear).
 Young, L.C. (1942) Generalized surfaces in the calculus of variations I, II. Ann. Math. 43: pp. 84103 CrossRef
 Young, L.C. (1951) Surfaces paramétriques généralisées. Bull. Soc. Math. France 79: pp. 5984
 Young, L.C. (1963) Some extremal questions for simplicial complexes, V: the relative area of a Klein bottle. Rend. Circ. Mat. Palermo 12: pp. 257274 CrossRef
 Title
 The mathematics of F. J. Almgren, Jr.
 Journal

The Journal of Geometric Analysis
Volume 8, Issue 5 , pp 681702
 Cover Date
 19980901
 DOI
 10.1007/BF02922665
 Print ISSN
 10506926
 Online ISSN
 1559002X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 Brian White ^{(1)}
 Author Affiliations

 1. Mathematics Department, Stanford University, 94305, Stanford, CA