Geometric Analysis


Geometric Analysis

The aim of this graduate-level text is to equip the reader with the basic tools and techniques needed for research in various areas of geometric analysis. Throughout, the main theme is to present the interaction of partial differential equations and differential geometry. More specifically, emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the underlying manifold and vice versa. For efficiency the author mainly restricts himself to the linear theory and only a rudimentary background in Riemannian geometry and partial differential equations is assumed. Originating from the author's own lectures, this book is an ideal introduction for graduate students, as well as a useful reference for experts in the field.


 Reviews:

"This monograph is a beautiful introduction to geometric analysis."
Frederic Robert, Mathematical Reviews

[A] F. Almgren Jr. , Some interior regularity theorems for minimal surfaces and an extension of Bernstein's Theorem, Ann. Math. 84(1966), 277–292.
[An] M. Anderson , The Dirichlet problem at infinity for manifolds of negative curvature, J. Diff. Geom. 18(1983), 701–721.
[B] S. Bochner , Vector fields and Ricci curvature, AMS Bull. 52(1946), 776–797.
[BC] R. Bishop and R. Crittenden , Geometry of manifolds, Academic Press, New York, 1964.
[BDG] E. Bombieri , E. De Giorgi , and E. Guisti , Minimal cones and the Bernstein problem, Invent. Math. 7(1969), 243–268.
[BGS] W. Ballmann , M. Gromov , and V. Schroeder , Manifolds of nonpositive curvature, vol. 61, Progr. Math, Birkhäuser, Berlin, 1985.
[Bm] E. Bombieri , Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimensions, Unpublished lecture notes (1970).
[Bo] A. Borbély , A note on the Dirichlet problem at infinity for manifolds of negative curvature, Proc. Amer. Math. Soc. 114(1992), 865–872.
[Br] R. Brooks , A relation between growth and the spectrum of the Laplacian, Math. Z. 178(1981), 501–508.
[Bu] P. Buser , On Cheeger's inequality λ1 ≥ h2/4, AMS Proc. Symp. Pure Math. 36(1980), 29–77.
[C] J. Cheeger , A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis, a symposium in honor of S. Bochner, Princeton University Press, Princeton, 1970, pp. 195–199.
[CCM] J. Cheeger , T. Colding , and W. Minicozzi , Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature, Geom. Func. Anal. 5(1995), 948–954.
[CG1] J. Cheeger and D. Gromoll , On the structure of complete manifolds of nonnegative curvature, Ann. Math. 92(1972), 413–443.
[Cg1] S.Y. Cheng , Eigenvalue comparison theorems and its geometric application, Math. Z. 143(1975), 279–297.
[CG2] J. Cheeger and D. Gromoll , The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom. 6(1971), 119–127.
[Cg2] S.Y. Cheng , Eigenfunctions and nodal sets, Comment. Math. Helv. 51(1976), 43–55.
[Cg3] S.Y. Cheng , Liouville theorem for harmonic maps, Proc. Symp. Pure Math. 36(1980), 147–151.
[CgY] S.Y. Cheng and S. T. Yau , Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 27, (1975), 333–354.
[CM1] T. Colding and W. Minicozzi , On function theory on spaces with a lower Ricci curvature bound, Math. Res. Lett. 3(1996), 241–246.
[CM2] T. Colding and W. Minicozzi , Generalized Liouville properties of manifolds, Math. Res. Lett. 3(1996), 723–729.
[CM3] T. Colding and W. Minicozzi , Harmonic functions with polynomial growth, J. Diff. Geom. 46(1997), 1–77.
[CM4] T. Colding and W. Minicozzi , Harmonic functions on manifolds, Ann. Math. 146(1997), 725–747.
[CM5] T. Colding and W. Minicozzi , Weyl type bounds for harmonic functions, Invent. Math. 131(1998), 257–298.
[CM6] T. Colding and W. Minicozzi , Liouville theorems for harmonic sections and applications, Comm. Pure Appl. Math. 51(1998), 113–138.
[Cn] R. Chen , Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. Amer. Math. Soc. 108(1990), 961–970.
[CnL] R. Chen and P. Li , On Poincaré type inequalities, Trans. AMS 349(1997), 1561–1585.
[CSZ] H. Cao , Y. Shen , and S. Zhu , The structure of stable minimal hypersurfaces in ℝn+1 , Math. Res. Let. 4(1997), 637–644.
[CV] S. Cohn-Vossen , Kürzeste Wege and Totalkrümmung auf Flächen, Compositio Math. 2(1935), 69–133.
[CW] H. I. Choi and A.N. Wang , A first eigenvalue estimate for minimal hypersurfaces, J. Diff. Geom. 18(1983), 559–562.
[CY] J. Cheeger and S. T. Yau , A lower bound for the heat kernel, Comm. Pure Appl. Math. 34(1981), 465–480.
[D] E.B. Davies , Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989.
[dCP] M. do Carmo and C.K. Peng , Stable complete minimal surfaces in ℝ3 are planes, Bull. AMS 1(1979), 903–906.
[De] E. De Giorgi , Una estensione del teorema di Bernstein, Ann. Scuola Nor. Sup. Pisa 19(1965), 79–85.
[E] J. Escobar , Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate, Comm. Pure Appl. Math. 43(1990), 857–883.
[EO] P. Eberlein and B. O'Neill , Visibility manifolds, Pacific J. Math. 46(1973), 45–110.
[F] C. Faber , Beweiss, dass unter allen homogenen Membrane von gleicher Fläche und gleicher Spannung die kreisförmige die tiefsten Grundton gibt, Sitzungsber.–Bayer. Akad. Wiss, Math.-Phys. München, 1923, 169–172.
[FC] D. Fischer-Colbrie , On complete minimal surfaces with finite Morse index in three manifolds, Invent. Math. 82(1985), 121–132.
[FCS] D. Fischer-Colbrie and R. Schoen , The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33(1980), 199–211.
[FF] H. Federer and W. Fleming , Normal and integral currents, Ann. Math. 72(1960), 458–520.
[Fl] W. Fleming , On the oriented plateau problem, Rend. Circ. Mat. Palerino 11(1962), 69–90.
[G1] A. Grigor'yan , On stochastically complete manifolds, Soviet Math. Dokl. 34(1987), 310–313.
[G2] A. Grigor'yan , On the dimension of spaces of harmonic functions, Math. Notes 48(1990), 1114–1118.
[G3] A. Grigor'yan , The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik 72(1992), 47–77.
[Ge] T. Gelander , Homotopy type and volume of locally symmetric manifolds, Duke Math. J. 124(2004), 459–515.
[GM] S. Gallot and D. Meyer , Operateur de courbure et Laplacien des formes differentielles d'une variété Riemannienne, J. Math. Pures Appl. (9) 54(1975), 259–274.
[Gu1] R. Gulliver , Index and total curvature of complete minimal surfaces, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), Proc. Sympos. Pure Math. 44, Amer. Math. Soc., Providence, RI., 1986, pp. 207–211.
[Gu2] R. Gulliver , Minimal surfaces of finite index in manifolds of positive scalar curvature, Lecture Notes in Mathematics: Calculus of variations and partial differential equations (Trento, 1986), vol. 1340, Springer, Berlin, 1988, pp. 115–122.
[HK] P. Hajtasz and P. Koskela , Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sr. I Math. 320(1995), 1211–1215.
[K] E. Krahn , Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94(1925), 97–100.
[Ka] A. Kasue , A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. Ecole. Norm. Sup. 21(1988), 593–622.
[KL] L. Karp and P. Li , The heat equation on complete Riemannian manifolds, http://math.uci.edu/pli/heat.pdf.
[KLZ] S. Kong , P. Li , and D. Zhou , Spectrum of the Laplacian on quaternionic Kähler manifolds, J. Diff. Geom. 78(2008), 295–332.
[L1] P. Li , A lower bound for the first eigenvalue for the Laplacian on compact manifolds, Indiana U. Math. J. 27(1979), 1013–1019.
[L2] P. Li , On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Scient. Ecole. Norm. Sup. 4, T 13(1980), 451–469.
[L3] P. Li , Poincaré inequalities on Riemannian manifolds, Seminar on Differential Geometry, Annals of Math. Studies. Edited by S. T. Yau , vol. 102, Princeton University Press, Princeton, 1982, pp. 73–83.
[L4] P. Li , Uniqueness of L1 solutions for the Laplace equation and the heat equation on Riemannian manifolds, J. Diff. Geom. 20(1984), 447–457.
[L5] P. Li , Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. Math. 124(1986), 1–21.
[L6] P. Li , Lecture notes on geometric analysis, Lecture Notes Series No. 6 - Research Institute of Mathematics and Global Analysis Research Center, Seoul National University, Seoul, 1993.
[L7] P. Li , Harmonic functions of linear growth on Kähler manifolds with nonnegative Ricci curvature, Math. Res. Lett. 2(1995), 79–94.
[L8] P. Li , Harmonic sections of polynomial growth, Math. Res. Lett. 4(1997), 35–44.
[L9] P. Li , Harmonic functions and applications to complete manifolds, XIV Escola de Geometria Diferencial: Em homenagem a Shiing-Shen Chern, IMPA, Rio de Janeiro, 2006.
[LoT] G. Liao and L. F. Tam , On the heat equation for harmonic maps from non-compact manifolds, Pacific J. Math. 153(1992), 129–145.
[LS] P. Li and R. Schoen , Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153(1984), 279–301.
[LT1] P. Li and L. F. Tam , Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set., Ann. Math. 125(1987), 171–207.
[LT2] P. Li and L. F. Tam , Symmetric Green's functions on complete manifolds, Amer. J. Math. 109(1987), 1129–1154.
[LT3] P. Li and L. F. Tam , Linear growth harmonic functions on a complete manifold, J. Diff. Geom. 29(1989), 421–425.
[LT4] P. Li and L. F. Tam , Complete surfaces with finite total curvature, J. Diff. Geom. 33(1991), 139–168.
[LT5] P. Li and L. F. Tam , The heat equation and harmonic maps of complete manifolds, Invent. Math. 105(1991), 1–46.
[LT6] P. Li and L. F. Tam , Harmonic functions and the structure of complete manifolds, J. Diff. Geom. 35(1992), 359–383.
[LW1] P. Li and J. Wang , Convex hull properties of harmonic maps, J. Diff. Geom. 48(1998), 497–530.
[LW2] P. Li and J. P. Wang , Mean value inequalities, Indiana Math. J. 48(1999), 1257–1273.
[LW3] P. Li and J. P. Wang , Counting massive sets and dimensions of harmonic functions, J. Diff. Geom. 53(1999), 237–278.
[LW4] P. Li and J. P. Wang , Minimal hypersurfaces with finite index, Math. Res. Lett. 9(2002), 95–103.
[LW5] P. Li and J. P. Wang , Complete manifolds with positive spectrum, J. Diff. Geom. 58(2001), 501–534.
[LW6] P. Li and J. Wang , Complete manifolds with positive spectrum, II., J. Diff. Geom. 62(2002), 143–162.
[LW7] P. Li and J. P. Wang , Stable minimal hypersurfaces in a nonnegatively curved manifold, J. Reine Angew. Math. (Crelles) 566(2004), 215–230.
[LW8] P. Li and J. Wang , Comparison theorem for Kähler manifolds and positivity of spectrum., J. Diff. Geom. 69(2005), 43–74.
[LY1] P. Li and S. T. Yau , Eigenvalues of a compact Riemannian manifold., AMS Proc. Symp. Pure Math. 36(1980), 205–239.
[LY2] P. Li and S. T. Yau , On the parabolic kernel of the Schrödinger operator, Acta Math. 156(1986), 153–201.
[Lz] A. Lichnerowicz , Géometrie des groupes de transformations, Dunod, Paris, 1958.
[M] B. Malgrange , Existence et approximation des solutions der équations aux dérivées partielles et des équations de convolution, Annales de l'Inst. Fourier 6(1955), 271–355.
[MS] J. H. Michael and L. Simon , Sobolev and mean-value inequalities on generalized submanifolds of ℝn , Comm. Pure Appl. Math. 26(1973), 361–379.
[NT] L. Ni and L. Tam , Kähler–Ricci flow and the Poincaré–Lelong equation, Comm. Anal. Geom. 12(2004), 111–141.
[O] M. Obata , Certain conditions for a Riemannian manifold to be isometric to the sphere, J. Math. Soc. Japan 14(1962), 333–340.
[R] R. Reilly , Applications of the Hessian operator in a Riemannian manifold, Indiana U. Math. J. 26(1977), 459–472.
[Ro] H. Royden , Harmonic functions on open Riemann surfaces, Trans. AMS 73(1952), 40–94.
[S] J. Simons , Minimal varieties in Riemannian manifolds, Ann. Math. 80(1964), 1–21.
[SC] L. Saloff-Coste , Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom. 36(1992), 417–450.
[Sh] Y. Shen , A Liouville theorem for harmonic maps, Amer. J. Math. 117(1995), 773–785.
[Si] L. Simon , Lectures on geometric measure theory, Proc. Centre for Mathematical Analysis, Australian National University, Canberra, 1984.
[St] E. Stein , Singular integrals and differentiability properties of functions, Princeton mathematical series no. 30, Princeton University Press, Princeton, 1970.
[STW] C. J. Sung , L. F. Tam , and J. P. Wang , Spaces of harmonic functions, J. London Math. Soc. 61(2000), 789–806.
[SY1] R. Schoen and S. T. Yau , Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature, Comm. Math. Helv. 39(1976), 333–341.
[SY2] R. Schoen and S. T. Yau , Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, vol. I, International Press., Cambridge.
[T] L. F. Tam , Liouville properties of harmonic maps, Math. Res. Lett. 2(1995), 719–735.
[V] N. Varopoulos , Hardy–Littlewood theory for semigroups, J. Funct. Anal. 63(1985), 240–260.
[WZ] J. Wang and L. Zhou , Gradient estimate for eigenforms of Hodge Laplacian, preprint.
[Y1] S. T. Yau , Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 27(1975), 201–228.
[Y2] S. T. Yau , Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana U. Math. J. 25(1976), 659–670.
[Y3] S. T. Yau , Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Scient. Ecole. Norm. Sup. 4(1985), 487–507.
[ZY] J. Q. Zhong and H. C. Yang , On the estimate of first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27(1984), 1265–1273.