Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

Year 2024, Volume: 17 Issue: 1, 290 - 305, 23.04.2024

Shihshu Walter Wei

https://doi.org/10.36890/iejg.1472310

Cited By: 1

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]).

Keywords

Poincáre inequality, topological end, Sobolev inequality, harmonic function, harmonic map, Ricci curvature

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).