B.S.: 1967, Stanford University-- Mathematics

M.S.: 1969, Stanford University-- Mathematics

Ph.D.: 1971, Stanford University-- Mathematics

*Dissertation:* "Regularity of Minimizing Surfaces of Prescribed
Mean Curvature".

*Advisor:* Robert Osserman

University of Minnesota, Assistant Professor, 1973-76

University of Minnesota, Associate Professor, 1976-85

University of Minnesota, Professor, 1985-

Institute for Advanced Study, Princeton, Spring 1980

Stanford University, Fall 1980

Stanford University, Fall 1983

Faculty of Sciences, University of Trento, Italy, March - May 1986

University of Bonn, Germany, May - August 1986

Max-Planck Institut for Mathematics, Bonn, Germany, Sept. 1986 - August 1987

Centre for Mathematical Analysis, Canberra, Australia, July -- December 1988

SFB 256, University of Bonn, Germany, January -- August 1989

SFB 256, University of Bonn, Germany, March -- July, 1993

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, Sept. 1999 -- Feb. 2000

University of Melbourne and Monash University, Australia, March -- July 2000

1. * Elliptic and Parabolic Methods in Geometry.* Ed. with Bennett
Chow, Silvio Levy and John Sullivan. A. K. Peters, Wellesley 1996.

2. *Differential Geometric Methods in the Control of Partial Differential
Equations.* Ed. with Walter Littman and Roberto Triggiani.
Contemporary Mathematics vol. 268: Amer. Math. Soc., Providence 2000.

1. "Nuclear magnetic resonance studies of diffusion of Al^{27}
in aluminum and aluminum alloys," * Acta Metallurgica ***13,**
701-708 (1965) (with T.G. Stoebe, T.O. Ogurtani and R.A. Huggins)

2. "On regularity of generalized surfaces of constant mean curvature,"
* Bull. Amer. Math. Soc.* **7,** 243-245 (1971).

3. "The Plateau problem for surfaces of prescribed mean curvature in a
cylinder," * Inventiones Math.* **13,** 169-178 (1971)
(with Joel Spruck).

4. "Existence theorems for parametric surfaces of prescribed mean
curvature," * Indiana University Math. J.* **22,** 445-472
(1972) (with Joel Spruck).

5. "Surfaces of constant mean curvature which have a simple projection,"
* Math. Z.* **129,** 95-107 (1972) (with Joel Spruck).

6. "Regularity of minimizing surfaces of prescribed mean curvature,"
* Annals of Math.* **97,** 275-305 (1973).

7. "The Plateau problem for surfaces of prescribed mean curvature in a
Riemannian manifold," * J. Differential Geometry* **8,**
317-330 (1973).

8. "Existence of surfaces with prescribed mean curvature vector,"
* Math. Z.* **131,** 117-140 (1973).

9. "On boundary branch points of minimizing surfaces," * Arch. Rat.
Mech. Anal.* **52,** 20-25 (1973) (with Frank David Lesley).

10. "A theory of branched immersions of surfaces," * Amer. J. Math.*
**95,** 750-812 (1973) (with R. Osserman and H. Royden).

11. "On the nonexistence of a hypersurface of prescribed mean curvature with
a given boundary," * Manuscripta Math.* **11,** 15-39 (1974).

12. 'On the variety of manifolds without conjugate points," * Trans. Amer.
Math. Soc. * **210,** 185-201 (1975).

13. "Branched immersions of surfaces and reduction of topological type, I:"
* Math. Z.* **145,** 267-288 (1975).

14. "Removability of singular points on surfaces of bounded mean curvature,"
* J. Differential Geometry * **11,** 345-350 (1976).

15. "Finiteness of the ramified set for branched immersions of surfaces,"
* Pacific J. Math. * **64,** 153-166 (1976).

16. "On embedded minimal surfaces," * Annals of Math. * **103, **
331-347 (1976), (with Joel Spruck).

16bis. "On embedded minimal surfaces, Correction," * Annals of Math. *
**109,** 407-412 (1979).

17. "Branched immersions of surfaces and reduction of topological type, II"
* Math. Ann. * **230,** 25-48 (1977).

18. "Representation of surfaces near a branched minimal surface, * Proc.
U.S.- Japan Seminar on Minimal Submanifolds and Geodesics," * Kaigai
Publications, Tokyo, 1978, pp. 31-41.

19. "Necessary conditions for submanifolds and currents with
prescribed mean curvature vector," * Annals of Mathematics Studies *
**103,** 225-242 (1983).

20. "The symmetry group of a curve preserves a plane," * Proc. Amer.
Math. Soc.* **84,** 408-411 (1982) (with Frank Morgan).

21. "Tori of prescribed mean curvature and the rotating drop,"
* Asterisque * **118,** 167-179 (1984).

22. "The structure of a stable minimal hypersurface near a singularity,"
* Proc. Symp. in Pure Math. * **44,** 213-237 (1986) (with Blaine
Lawson).

23. "Index and total curvature of complete minimal surfaces," * Proc.
Symp. in Pure Math. * **44,** 207-211 (1986).

24. "Planes without conjugate points," * J. Diff. Geom. * **22, **
43-47 (1985) (with Leon Green).

25. "Harmonic maps which solve a free boundary problem," * J. Reine
Angewandte Math. * **381,** 61-89 (1987) (with Jürgen Jost).

26. "Boundary configurations spanning infinitely many minimal surfaces,"
* Manuscripta Math. * **54,** 323-347 (1986) (with Stefan
Hildebrandt).

27. "Least-area surfaces can have excess triple points," * Topology *
**26,** 61-89 (1987) (with Peter Scott).

28. "Minimal surfaces of finite index in manifolds of positive scalar
curvature," Springer Verlag * Lecture Notes Math. * **1340,**
115-122 (1988).

29. "Isoperimetric problems having continua of solutions," Springer-Verlag
* Lect. Notes Math.* **1357,** 256-263 (1988).

30. "The rate of convergence of a harmonic map at a singular point,"
* Math. Annalen * **283,** 539-549 (1989) (with Brian White).

31. "Minimizing p-harmonic maps into spheres" * J. Reine Angewandte
Math. * **401** (1989), 82-100 (with Jean-Michel Coron).

32. "On false branch points of incompressible branched immersions"
* Manuscripta Math. * **63** (1989), 293-302 (with Friedrich Tomi).

33. "Convergence of minimal submanifolds to a singular variety," *
Progress in Nonlin. DE's and Applics.* vol. **4**: Variational
Methods, 297-308. Birkhauser, Boston 1990.

34. "A minimal surface with an atypical boundary branch point," pp. 211-228
of * Differential Geometry: a Symposium in honor of Manfredo P. do
Carmo, * B. Lawson and K. Tenenblat, eds., Longman, Harlow 1991.

35. "The sharp isoperimetric inequality for minimal surfaces with radially
connected boundary in hyperbolic space" (with Jaigyoung Choe),
* Inventiones Math. * **109** (1992), 495-503.

36. "Isoperimetric inequalities on minimal submanifolds of space forms"
(with Jaigyoung Choe), * Manuscripta Math. * **77** (1992),
169-189.

37. "A conjectured heat flow problem" (with N. B. Willms), * SIAM Review
* **37** (1995), 100--104.

38. "Aleksandrov reflection and nonlinear evolution equations, I: The
n-sphere and n-ball" (with Bennett Chow), * Calculus of Variations and
Partial Differential Equations * **4** (1996), 249--264.

39. "Higher eigenvalue estimate on manifold" (with Bang-Ok Kim), * Comm.
Korean Math. Soc. * **13** (1998), 579--587.

40. "Chord uniqueness and controllability: the view from the boundary, I"
(with Walter Littman), * Contemporary Mathematics * **268**
(2000), 145--176.

41. "Aleksandrov reflection and geometric evolution of hypersurfaces"
(with Bennett Chow), * Comm. Anal. Geom.* **9** (2001), 261--280.

42. "Sharp growth rate for generalized solutions evolving by mean
curvature plus a forcing term" (with Yonghoi Koo),
* Journal für die Reine und Angewandte Math. * **538**
(2001), 1--24.

43. "Boundary control of PDEs via curvature flows: the view from the
boundary, II" (with Santiago Betelú and Walter Littman).
* Applied Mathematics and Optimization * **46** (2002), 167-178
(volume in memory of J.L.Lions.)

44. "Embedded minimal surfaces and total curvature of curves in
a manifold" (with Jaigyoung Choe), * Math Research Letters *
**10** (2003), 343-362.

45. "Analysis of a variational approach to progressive lens
design" (with Jing Wang and Fadil Santosa), * SIAM J. Appl. Math. *
**64** (2003), 277-296.

46. "The case for differential geometry in the control of single and
coupled PDEs: The structural acoustic chamber" (with Irena
Lasiecka, Walter Littman and Roberto Triggiani), * IMA Volumes
in Mathematics and its Applications, * vol. **137,** 73--182
(2004).

47. "Density estimates for minimal surfaces and surfaces flowing by
mean curvature." In Proceedings of Workshop on Geometric
Evolution Equations, NCTS, Taiwan (July 2002). * Contemporary
Mathematics ***367,** 129-140 (2005).

48. "Area density and regularity for soap film-like surfaces
spanning graphs" (with Sumio Yamada), *Math. Z.* **253**
(2006), 315--331.

49. "Total Curvature of Graphs in Space," *Quarterly
Journal of Pure and Applied Mathematics* **3** (2007), 773—783.

50. "Examples of hypersurfaces flowing by curvature in a Riemannian
manifold" (with Guoyi Xu), *Comm. Anal. Geom.* **17** (2009),
701--720.

51. "Regularity of soap film-like surfaces spanning graphs in a
Riemannian manifold" (with Sung-ho Park, Juncheol Pyo, and Keomkyo
Seo), to appear in *Journal of Korean Malth Society*.

52. "Total Curvature of graphs after Milnor and Euler" (with Sumio Yamada), ArXiv:1101.2305. To appear in Pacific Journal of Mathematics.

53. "Fiber evolution in the heat flow of harmonic maps." NATO ASI Series
C, vol. **332**: *Nematics,* 173-178. Kluwer, Dordrecht 1991.

54. "On convergence rates of harmonic maps near points of discontinuity"
(with Brian White), in H. Brezis & J.-L. Lions, eds.,
*Nonlinear Partial Differential Equations and their applications,*
vol. **X,** 216-228. Wiley, New York 1991.

55. "Harmonic mappings, symmetry and computer graphics," MSRI Publ.
vol. **17**: *Geometric Analysis and Computer Graphics,* 79-88.
Springer-Verlag, New York 1991.

56. "The heat-flow method in contact geometry." *Proceedings of
Symposium on Geometric Analysis and Applications* (2000), Proceedings
of the Centre for Mathematics and its Applications, vol. **39,**
106--117, Australian National University, 2001.

57. "The use of geometric tools in the boundary control of partial
differential equations" (with Walter Littman).
*Progress in Analysis: Proceedings of the 3rd ISAAC Congress,*
Berlin 2001, Vol. **1,** 493--497 (H. G. W. Begehr, R. P. Gilbert,
M. W. Wong, eds.) World Scientific Publishing Co., Inc., River
Edge, NJ, 2003.

58. "An index theorem for H-surfaces in R^{3} of arbitrary genus
and with branch points" (with Reinhold Böhme), preprint, 24 pp.

59. "Aleksandrov reflection and nonlinear evolution equations, II: manifolds with reflectional symmetry" (with Bennett Chow), in preparation.

60. "The Aleksandrov reflection principle for self-intersecting hypersurfaces," in preparation.

61. "Asymptotic spectrum of manifolds of increasing topological type" (with Gianni dal Maso and Umberto Mosco), preprint, 92 pp.

62. "Boundary control of wave equations in high dimensions via harmonic mean-curvature flow" (with Walter Littman), in preparation.

MSRI Workshop on Differential Geometry, Calculus of Variations and Computer Graphics: "Symmetry and asymmetry of minimizing harmonic maps." May 24, 1988.

Ecole Normale Superieure Workshop on Variational Problems: "A notion of weak convergence of minimal varieties to a singular variety." June 17, 1988.

Oberwolfach Conference on the Calculus of Variations: "Variational convergence of minimal submanifolds to a singular variety." June 20, 1988.

Centre for Mathematical Analysis Colloquium, Canberra: "Maps of minimum energy into spheres". Sept. 8, 1988.

CMA Partial Differential Equations Seminar: "Boundary branch points of minimal surfaces". Oct. 10, 1988.

CMA Geometry Seminar: "Stability of minimal surfaces in manifolds of nonnegative scalar curvature". November 15, 1988.

CNRS Laboratory at Orsay: "The geometry of fibers of a harmonic map." April 20, 1989.

Oberseminar für Analysis, Universität Bonn: "Representation of Dirichlet forms and weak convergence of manifolds of unbounded topological type." June 15, 1989.

NATO Advanced Research Workshop on Defects, Singularities and Patterns in Liquid Crystals (Orsay): "The geometry of fibers for the heat flow of harmonic maps." May 29, 1990.

Geometry Colloquium, Universitat Heidelberg: "The sharp isoperimetric inequality for minimal surfaces in hyperbolic space." July 1, 1991.

Oberseminar fur Analysis, Universitat Bonn: "The sharp isoperimetric inequality for radially connected minimal surfaces in hyperbolic space." July 11, 1991.

CIRM Conference on Surface Tension, Trento: "Isoperimetric inequalities on minimal surfaces". July 23, 1992.

Oberwolfach Conference on Calculus of Variations: "The Fredholm index for branched H-surfaces of arbitrary genus". July 29, 1992.

GARC Conference on Differential Geometry, Seoul National University, Korea: "Connected sums of complete immersed minimal surfaces," Feb. 15, 1993; "Building blocks for complete immersed minimal surfaces," Feb. 17, 1993.

SFB/IAM Conference on Geometry and the Calculus of Variations, Bonn: "Connected sums of complete immersed minimal surfaces". May 14, 1993.

Lecture series at the Graduiertenkolleg, University of Bonn: "Singularities of harmonic maps." June 16 to July 14, 1993.

Analysis seminar, E.T.H. Zurich: "Connected sums of complete immersed minimal surfaces". June 1, 1993.

Mathematics Colloquium, R.W.T.H. Aachen: "Connected sums of complete minimal surfaces." July 13, 1993.

Mathematics Colloquium, University of Tübingen: "Connected sums of complete minimal surfaces." June 11, 1993.

Mathematics Colloquium, University of the Saarland, Saarbrücken: "Connected sums of complete minimal surfaces." June 25, 1993.

International Conference on Advances in Geometric Analysis and Continuum Mechanics, Stanford University: "Connected Sums of Complete Minimal Surfaces." August 3, 1993.

ICTP Conference on Composite Media and Homogenization Theory, Trieste: invited as a main speaker, September 20-October 1, 1993.

Mathematics Colloquium, Rice University: "Aleksandrov's method and hypersurfaces expanding by functions of curvature." Nov. 10, 1994.

Conference in honor of Robert Osserman, Stanford University: "Aleksandrov reflection and self-interesecting surfaces expanding by functions of curvature." May 6, 1995.

Mount Stromlo Miniconference on Mathematical Analysis, Canberra, Australia: "Sharp Growth Rate for Generalized Solutions Evolving by Mean Curvature plus a Forcing Term." September 3, 1998.

International Conference on Partial Differential Equations and Related Topics, Mission Beach, Queensland, Australia: "The Aleksandrov reflection method for hypersurfaces expanding by a function of curvatures." Mission Beach, Australia. Sept. 7, 1998.

AMS Summer Conference on Differential Geometric Methods in the Control of Partial Differential Equations: Boulder, Colorado. "Chord uniqueness and controllability: the view from the boundary." June 28, 1999.

Oberseminar Analysis, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. "The Aleksandrov reflection method for hypersurfaces expanding by a function of curvatures." October 19, 1999.

Oberseminar Analysis, University of Bonn, Germany. "The Aleksandrov reflection method for hypersurfaces expanding by a function of curvatures." October 21, 1999.

Analysis Seminar, Mathematisches Forschungsinstitut, ETH Zuerich, Switzerland. "The Aleksandrov reflection method for hypersurfaces expanding by a function of curvatures." November 16, 1999.

Arbeitsgemeinschaft Geometrie, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. "The sharp rate of fattening for viscosity solutions of hypersurfaces flowing by mean curvature." December 13, 1999.

Mathematics Colloquium, University of Heidelberg, Germany. "Aleksandrov reflection for hypersurfaces expanding by curvature." December 17, 1999.

Analysis Seminar, University of Rome II, Italy: "Aleksandrov reflection for hypersurfaces expanding by a function of curvatures." Jan. 24, 2000.

Geometry seminar, University of Rome II, Italy: "Sharp growth rate for viscosity solutions of Flow by Curvature ." January 26, 2000.

Mathematics Colloquium, University of Leipzig, Germany. "The shape of a surface which expands according to its curvature." February 2, 2000.

Geometry/Topology Seminar, University of Melbourne, Australia. "Rate of fattening for hypersurfaces flowing by mean curvature." March 15, 2000.

Geometric Analysis Seminar, Monash University, Australia. "Rate of Fattening for Hypersurfaces Flowing by Mean Curvature: Beyond Self-similarity." March 30, 2000.

Mathematics Colloquium, Monash University, Australia. "The shape of a surface which expands according to its curvatures." April 6, 2000.

Mathematics Colloquium, Australian National University, Canberra. "Control of wave equations via geometry: How long does it take to calm the lake?" May 18, 2000.

Analysis/Geometry Seminar, Monash University, Australia. "The Heat-Flow Method for Contact Forms." June 1, 2000.

Mathematics Colloquium, University of Melbourne, Australia. "Contact structures: Twist Locally, Flow Globally." June 20, 2000.

National Research Symposium on Geometric Analysis and Applications, Canberra, Australia. "The heat-flow method in contact geometry." June 27, 2000.

Plenary address to Annual Meeting of Australian Mathematical Society, Brisbane, Australia. "Surfaces which move according to curvatures." July 4, 2000.

Shoemaker Lectures, University of Toledo: "Surfaces Which Move According to Curvatures," October 25, 2000; "Maximum Principles for Viscosity Solutions," October 26, 2000; "The heat flow method in contact geometry (Twist Locally, Flow Globally)," October 27, 2000.

Institute for Mathematics and its Applications Summer Program 2001: Geometric Methods in Inverse Problems and PDE Control. "Boundary control of wave equations via geometry", July 19, 2001.

Pacific Inistitute of Mathematical Sciences Thematic Programme 2001 on Nonlinear Partial Differential Equations, Session 5: Geometric PDEs. "Embedded minimal surfaces and total curvature of curves in a manifold", August 10, 2001.

NCTS workshop on Geometric Evolution Equations, National Tsin-Hua University, Taiwan. "A density estimate for minimal surfaces", July 23, 2002.

University of Alabama at Birmingham Math Colloquium: "Curves of Small Curvature and Embedded Minimal Surfaces", October 24, 2003.

Seoul National University, Korea, Differential Geometry Seminar: "Regularity of Soap Film-Like Surfaces and Total Curvature of Graphs", January 8, 2004.

Chung-Ang University, Korea, Math Colloquium: "Curves of Small Curvature and Embedded Minimal Surfaces", January 16, 2004.

Mathematics Junior Colloquium,
University of Minnesota: "How to Build Minimal Surfaces",
March 1, 2005.

Workshop on Geometry & Topology, University of Minnesota:
"Chern-Gauss-Bonnet, Chern-Lashof and minimal submanifolds",
March 16, 2005.

Conference on Geometry and PDE, Stanford University: "Total
curvature and isotopy of graphs in R^3", September 15, 2005.

ECNU Workshop on Nonlinear PDEs, East China Normal University,
Shanghai, China: "Curvature of boundaries and density estimates for
minimal submanifolds", June 11, 2006.

Colloquium, Mathematics Department, Beijing University, Beijing,
China: "Curvature of boundaries and density estimates for minimal
submanifolds", June 20, 2006.

Colloquium, Brown University, Providence, Rhode Island: ``Total
curvature of curves and of graphs" (April 6, 2007).

Invited speaker for 2007 Bloomington Geometry Workshop, Indiana
University: ``Total curvature of curves and of graphs" (April 14,
2007).

Invited speaker for KIAS Conference on Geometric Analysis, Korea
Institute of Advanced Study: ``Total curvature of curves and of
graphs" (September 10, 2007).

Chinese University of Hong Kong: ``Total curvature and isotopy of
graphs in $R^3$" (January 4, 2008)

Colloquium, Notre Dame University: ``Total curvature and isotopy
of smooth graphs in R^3" (January 16, 2008).

Geometry seminar, Notre Dame University: ``Total curvature and
isotopy of continuous graphs" (January 17, 2008).

Conference on "Geometric Flows in Mathematics and Theoretical
Physics" at De Giorgi Center, Pisa (June 25, 2009)

Workshop on ``Mean Curvature Flow", University of Valencia, Spain
(September 2, 2010)

Appointed as Co-Organizer of a special year in Nonlinear Partial Differential Equations at the Max Planck Institute for Mathematics, Bonn, Germany (September 1986 -- August 1987)

Invited as University Fellow to visit the Centre for Mathematical Analysis at the Australian National University (July -- Dec. 1988)

Invited by Sonderforschungsbereich 256: Nichtlineare Partielle Differentialgleichungen to spend 8 months at the University of Bonn (Jan. -- Aug. 1989)

Invited speaker at NATO Advanced Research Workshop, May 1990 at Orsay, France.

Co-organizer of A.M.S. Special Session on the Geometry of Equilibrium Configurations, Fargo, ND (Oct. 25 -- 26, 1991)

Invited by SFB 256: Nonlinear Partial Differential Equations to spend 4 months at the University of Bonn (April -- July 1993).

Invited to lecture to the Graduiertenkolleg (consortium of German Ph.D.-granting math departments) on "Singularities of Harmonic Maps" (June 16 -- July 14, 1993)

Co-organizer of Geometry Center Workshop on "Elliptic and Parabolic Methods in Geometry" (May 23 -- 27, 1994).

Co-organizer of conference on "Pure, Industrial and Applied Mathematics: Strength through Connections" in celebration of Avner Friedman's 65th birthday, Minneapolis, Minnesota (May 1 -- 3, 1998).

Co-organizer of conference on "Nonlinear Partial Differential Equations and Applications to Materials", Minneapolis, Minnesota (April 30 -- May 2, 1999).

Co-organizer of AMS Summer Conference on "Differential Geometric Methods in the Control of Partial Differential Equations", University of Colorado, Boulder (June 27 -- July 1, 1999).

Invited by Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany to spend six months as research visitor (August 1999--Feb. 2000).

Invited by University of Melbourne, Australia as Principal Fellow in the Department of Maths and Stats, March 1--July 31, 2000.

Chair of organizing committee for First Yamabe Memorial Symposium: "Geometry and Analysis", Minneapolis, Minnesota (September 20 -- 22, 2002).

Chair of organizing committee for Second Yamabe Memorial Symposium: "Geometry and Physics", Minneapolis, Minnesota (September 17 -- 19, 2004).

Chair of organizing committee for Third Yamabe Memorial Symposium: "Geometry and Symplectic Topology", Minneapolis, Minnesota (September 22 -- 24, 2006).

Gene Johnson, Ph. D. 1982.

Shimon Silman, M.S. 1988.

Heayong Shin, Ph. D. 1991.

Annette Kavanaugh, M.S. 1993.

Preston Nichols, Ph. D. 1995.

Yonghoi Koo, Ph. D. 1998.

Jing Wang, Ph. D. 2001.

Michael Galbraith, Ph. D. 2004.

Institute for Mathematics and its Applications:
Associate Director, 1994--98, 2001--2002.

The contents of this page have not been reviewed or approved by the

University of Minnesota.