Research Article
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Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

Year 2024, , 290 - 305, 23.04.2024
https://doi.org/10.36890/iejg.1472310

Abstract

We connect the Poincaré inequality with the Sobolev inequality on Riemannian manifold in a family of integral inequalities $(1.5)$. For these continuum of inequalities, we obtain topological structure theorems of manifolds generalizing previous unification theorems in both intrinsic and extrinsic settings ([33]). Manifolds which admit any of these integral inequalities are nonparabolic, affect topology, geometry, analysis, and admit nonconstant bounded harmonic functions of finite energy. As a consequence, we have proven a Conjecture of Schoen-Yau ([27, p.74]) to be true in dimension two with hypotheses weaker than that used in [1] and [33]$($ which were weaker than the hypotheses set in the conjecture, $($ cf. Remark 1.5$)$. In the same philosophy and spirit as in ([31]), we prove that if $M$ is a complete $n$-manifold, satisfying $\operatorname{(i)}$ the volume growth condition $(1.1)$, $\operatorname{(ii)}$ Liouville Theorem for harmonic functions, and either $\operatorname{(v)}$ a generalized Poincaré- Sobolev inequality $(1.5)$, or $\operatorname{(vi)}$ a general integral inequality $(1.6)$, and Liouville Theorem for harmonic map $u : M \to K$ with $\operatorname{Sec}^K \le 0$, then $(1)$ $M$ has only one end and $(2)$ there is no nontrivial homomorphism from fundamental group $\pi_1(\partial D)$ into $\pi_1 (K)$ as stated in Theorem $1.5$. Some applications in geometry $(\S 3)$, geometric analysis $(\S 4)$, nonlinear partial differential systems $(\S 5)$, integral inequalities on complete noncompact manifolds $(\S 6)$ are made $($cf. e.g., Theorems $3.1$, $4.1$, $5.1$, and $6.1)$. Whereas we made the first study in ([29, 32]) on how the existence of an essential positive supersolution of a second order partial differential systems $Q(u)=0$ on a Riemannian manifold $M$, (by which we mean a $C^2$ function $v \ge 0$ on $M$ that is positive almost everywhere on $M$, and that satisfies $Q(v)=\operatorname{div}(A(x,v,\nabla v)\nabla v)+b(x,v,\nabla v)v\leq 0\quad (5.1)\, $) affects topology, geometry, analysis and variational problems on the manifold $M$. Whereas we generate the work in [35], under $p$-parabolic stable condition without assuming the $p$-th volume growth condition $\lim _{r \to \infty} r^{-p}\operatorname{Vol}(B_r) =0$. The techniques, concepts, and results employed in this paper can be combined with those of essential positive supersolutions of degenerate nonlinear partial differential systems $($cf. for example, Theorems 5.1 - 5.5, 6.1, etc.$)\, $ generalizing previous work in [32, 4.11], which in term recaptures the work of Schoen-Simon-Yau ([25, Theorem 2]). The combined techniques, concepts and method of [32] and [35] can also be used in other new manifolds we found by an extrinsic average variational method ([34]).

References

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  • [2] Burstall, F.E.: Harmonic maps of finite energy from non-compact manifolds. J. London Math. Soc. 30, 361–370(1984).
  • [3] Cao, H.D., Shen, Y. and Zhu, S.H.: The structure of stable minimal hypersurfaces in Rn+1. Math. Res. Lett.4 no. 5, 637–644 (1997).
  • [4] Carron, G.: Inégalités isopérimétriques de Faber-Krahn et conséquences. Actes de la Table Ronde de Géométrie Différentielle (Luminy 1992), Sémin. Cong.,1. Soc. Math. France, Paris, 205–232 (1996).
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  • [6] Cheeger, J. and Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geom. 6 119–128 (1971).
  • [7] Chen, B.Y. and Wei, S.W.: Sharp growth estimates for warping functions in multiply warped product manifolds. J. Geom. Symmetry Phys. 52, 27-46 (2019).
  • [8] Chen, B.Y. and Wei, S.W.: Riemannian submanifolds with concircular canonical field. Bull. Korean Math. Soc. 56 no. 6, 1525-1537 (2019).
  • [9] Cheng, S. Y. and Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28, 333-354 (1975).
  • [10] Carmo, M. do and Peng, C.K.: Stable complete minimal surfaces in R3 are planes. Bull. Amer. Math. Soc. 1, 903–906 (1979).
  • [11] Fischer-Colbrie, D. and Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative sector curvature. Comm. Pure Appl. Math. 33, 199–211 (1980).
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  • [13] Gilbarg, D. and Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1977).
  • [14] Greene, R.E. and Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27, 265–298 (1974).
  • [15] Gromoll, D. and Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. 90, 75–90 (1969).
  • [16] Hoffman, D. A., Ossorman, R. and Schoen, R.: On the Gauss map of complete surfaces of constant mean curvature in R3 and R4. Comment. Math. Helv. 57, 519–531 (1982).
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  • [18] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32 no. 1, 13–72 (1957).
  • [19] Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179, 535–554 (1982).
  • [20] Michael, J. and Simon, L. M.: Sobolev and Mean-value inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math. 26, 361–379 (1973).
  • [21] Milnor, J.: A note on curvature and the fundamental group. J. Diff. Geom. 2, 1–7(1968).
  • [22] Pogorelov, A. V. : On the stability of minimal surfaces. Dokl. Akad. Nauk SSSR 260, 293–295 (1981), Zbl. 495.53005 English transl.: Soviet Math. Dokl. 24, (1981).
  • [23] Rourke, C. P. and Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Springer-Verlag , viii + 123 pp. (1972).
  • [24] Sario, L., Schiffer, M. and Glasner, M.: The span and principal fucntions in Riemannian spaces. J. Analyse Math. 15, 115–134 (1965).
  • [25] Schoen, R., Simon, L. and Yau, S. T.: Curvature estimates for minimal hypersurfaces. Acta Math 134, 275–288 (1975).
  • [26] Schoen, R. and Yau, S. T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature. Comment. Math. Helv. 51 no. 3, 333–341 (1976).
  • [27] Schoen, R. and Yau, S.T.: Lectures on Differential Geometry. International Press Co, Hong Kong, Boston.(1994).
  • [28] Varopoulos, N. T.: Potential theory and diffusion on Riemannian manifolds. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II. Wadswarth Math. Ser. 821–837 (1983).
  • [29] Wei, S. W.: Essential positive supersolutions of nonlinear degenerate partial differential equations on Riemannian manifolds. Advances in Geometric Analysis and Continuum Mechanics, International Press., 275–287 (1995).
  • [30] Wei, S. W.: Representing homotopy groups and spaces of maps by p-harmonic maps. Indiana Univ. Math. J. 47 no. 2, 625- 670 (1998).
  • [31] Wei, S.W.: The balance between existence theorems and nonexistence theorems in differential goemetry. Tamkang J. of Math. 32(1), 61–88 (2001).
  • [32] Wei, S.W.: Nonlinear partial differential systems on Riemannian manifolds with their geometric applications. J. Geom. Analysis. 12 no. 1, 147–182 (2002).
  • [33] Wei, S. W.: The Structure of Complete Minimal Submanifolds in Complete Manifolds of Nonpositive Curvature. Houston J. of Math. 29 no. 3, 675-689 (2003).
  • [34] Wei, S. W.: An Extrinsic average variational methods, Φ(i)-harmonic maps and Φ(i)-SSU manifolds, i = 1, 2, 3. the Romanian Journal of Mathematics and Computer Science, 13 Issue 2, 100-124 (2023).
  • [35] Wei, S.W., Li, J. F. and Wu, L. N.: p-Parabolicity and a generalized Bochner’s method with applications. to appear in La Matematica. Official Journal of the Association for Women in Mathematics (2024).
  • [36] Yau, S. T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975).
  • [37] Yau, S. T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Scient. Ec. Norm. Sup. 8, 487–507 (1975).
Year 2024, , 290 - 305, 23.04.2024
https://doi.org/10.36890/iejg.1472310

Abstract

References

  • [1] Benjamini, I. and Cao, J. G.: Examples of simply-connected Liouville manifolds with positive spectrum. Differential.Geom.Appl. 6 no. 1, 31–50 (1996).
  • [2] Burstall, F.E.: Harmonic maps of finite energy from non-compact manifolds. J. London Math. Soc. 30, 361–370(1984).
  • [3] Cao, H.D., Shen, Y. and Zhu, S.H.: The structure of stable minimal hypersurfaces in Rn+1. Math. Res. Lett.4 no. 5, 637–644 (1997).
  • [4] Carron, G.: Inégalités isopérimétriques de Faber-Krahn et conséquences. Actes de la Table Ronde de Géométrie Différentielle (Luminy 1992), Sémin. Cong.,1. Soc. Math. France, Paris, 205–232 (1996).
  • [5] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis (Papers dedicated to Salomon Bochner, 1969). Princeton Univ. Press, 195–199 (1970).
  • [6] Cheeger, J. and Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geom. 6 119–128 (1971).
  • [7] Chen, B.Y. and Wei, S.W.: Sharp growth estimates for warping functions in multiply warped product manifolds. J. Geom. Symmetry Phys. 52, 27-46 (2019).
  • [8] Chen, B.Y. and Wei, S.W.: Riemannian submanifolds with concircular canonical field. Bull. Korean Math. Soc. 56 no. 6, 1525-1537 (2019).
  • [9] Cheng, S. Y. and Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28, 333-354 (1975).
  • [10] Carmo, M. do and Peng, C.K.: Stable complete minimal surfaces in R3 are planes. Bull. Amer. Math. Soc. 1, 903–906 (1979).
  • [11] Fischer-Colbrie, D. and Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative sector curvature. Comm. Pure Appl. Math. 33, 199–211 (1980).
  • [12] Galloway, G.: A generalization of Cheeger and Gromoll splitting theorem. Arch. Math (Basel). 47 no. 4, 372–375 (1986).
  • [13] Gilbarg, D. and Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1977).
  • [14] Greene, R.E. and Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27, 265–298 (1974).
  • [15] Gromoll, D. and Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. 90, 75–90 (1969).
  • [16] Hoffman, D. A., Ossorman, R. and Schoen, R.: On the Gauss map of complete surfaces of constant mean curvature in R3 and R4. Comment. Math. Helv. 57, 519–531 (1982).
  • [17] Hoffman, D. and Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian manifolds. Comm. Pure and Applied Math. XXVII, 715–727 (1989).
  • [18] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32 no. 1, 13–72 (1957).
  • [19] Karp, L.: Subharmonic functions on real and complex manifolds. Math. Z. 179, 535–554 (1982).
  • [20] Michael, J. and Simon, L. M.: Sobolev and Mean-value inequalities on generalized submanifolds of Rn. Comm. Pure Appl. Math. 26, 361–379 (1973).
  • [21] Milnor, J.: A note on curvature and the fundamental group. J. Diff. Geom. 2, 1–7(1968).
  • [22] Pogorelov, A. V. : On the stability of minimal surfaces. Dokl. Akad. Nauk SSSR 260, 293–295 (1981), Zbl. 495.53005 English transl.: Soviet Math. Dokl. 24, (1981).
  • [23] Rourke, C. P. and Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Springer-Verlag , viii + 123 pp. (1972).
  • [24] Sario, L., Schiffer, M. and Glasner, M.: The span and principal fucntions in Riemannian spaces. J. Analyse Math. 15, 115–134 (1965).
  • [25] Schoen, R., Simon, L. and Yau, S. T.: Curvature estimates for minimal hypersurfaces. Acta Math 134, 275–288 (1975).
  • [26] Schoen, R. and Yau, S. T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature. Comment. Math. Helv. 51 no. 3, 333–341 (1976).
  • [27] Schoen, R. and Yau, S.T.: Lectures on Differential Geometry. International Press Co, Hong Kong, Boston.(1994).
  • [28] Varopoulos, N. T.: Potential theory and diffusion on Riemannian manifolds. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II. Wadswarth Math. Ser. 821–837 (1983).
  • [29] Wei, S. W.: Essential positive supersolutions of nonlinear degenerate partial differential equations on Riemannian manifolds. Advances in Geometric Analysis and Continuum Mechanics, International Press., 275–287 (1995).
  • [30] Wei, S. W.: Representing homotopy groups and spaces of maps by p-harmonic maps. Indiana Univ. Math. J. 47 no. 2, 625- 670 (1998).
  • [31] Wei, S.W.: The balance between existence theorems and nonexistence theorems in differential goemetry. Tamkang J. of Math. 32(1), 61–88 (2001).
  • [32] Wei, S.W.: Nonlinear partial differential systems on Riemannian manifolds with their geometric applications. J. Geom. Analysis. 12 no. 1, 147–182 (2002).
  • [33] Wei, S. W.: The Structure of Complete Minimal Submanifolds in Complete Manifolds of Nonpositive Curvature. Houston J. of Math. 29 no. 3, 675-689 (2003).
  • [34] Wei, S. W.: An Extrinsic average variational methods, Φ(i)-harmonic maps and Φ(i)-SSU manifolds, i = 1, 2, 3. the Romanian Journal of Mathematics and Computer Science, 13 Issue 2, 100-124 (2023).
  • [35] Wei, S.W., Li, J. F. and Wu, L. N.: p-Parabolicity and a generalized Bochner’s method with applications. to appear in La Matematica. Official Journal of the Association for Women in Mathematics (2024).
  • [36] Yau, S. T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975).
  • [37] Yau, S. T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Scient. Ec. Norm. Sup. 8, 487–507 (1975).
There are 37 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Shihshu Walter Wei This is me 0009-0006-3710-6038

Early Pub Date April 22, 2024
Publication Date April 23, 2024
Submission Date March 20, 2024
Acceptance Date April 22, 2024
Published in Issue Year 2024

Cite

APA Wei, S. W. (2024). Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds. International Electronic Journal of Geometry, 17(1), 290-305. https://doi.org/10.36890/iejg.1472310
AMA Wei SW. Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds. Int. Electron. J. Geom. April 2024;17(1):290-305. doi:10.36890/iejg.1472310
Chicago Wei, Shihshu Walter. “Connecting Poincaré Inequality With Sobolev Inequalities on Riemannian Manifolds”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 290-305. https://doi.org/10.36890/iejg.1472310.
EndNote Wei SW (April 1, 2024) Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds. International Electronic Journal of Geometry 17 1 290–305.
IEEE S. W. Wei, “Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 290–305, 2024, doi: 10.36890/iejg.1472310.
ISNAD Wei, Shihshu Walter. “Connecting Poincaré Inequality With Sobolev Inequalities on Riemannian Manifolds”. International Electronic Journal of Geometry 17/1 (April 2024), 290-305. https://doi.org/10.36890/iejg.1472310.
JAMA Wei SW. Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds. Int. Electron. J. Geom. 2024;17:290–305.
MLA Wei, Shihshu Walter. “Connecting Poincaré Inequality With Sobolev Inequalities on Riemannian Manifolds”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 290-05, doi:10.36890/iejg.1472310.
Vancouver Wei SW. Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds. Int. Electron. J. Geom. 2024;17(1):290-305.