## Sharp local isoperimetric inequalities involving the scalar curvature.(English)Zbl 1067.53026

This paper provides sharp local isoperimetric inequalities on complete and compact Riemannian manifolds. The first result is the following:
Theorem 1. {Let $$(M, g)$$ be a complete Riemannian manifold of dimension $$n\geq 2$$ and let $$x\in M.$$ Assume that $$S_g(x)<n(n-1)K_0$$ for some $$K_0\in \mathbf R$$ (where $$S_g$$ denotes the scalar curvature of $$(M, g)$$). Then there exists $$r_x>0$$ such that for any $$\Omega$$ contained in the geogesic ball of center $$x$$ and radius $$r_x,$$ $| \partial\Omega| _g>| \partial B| _{g_0}$ where $$B$$ is a ball of volume $$| \Omega| _g$$ in the model space $$(M_0, g_0)$$ of constant sectional curvature $$K_0.$$}
The proof of this theorem is based on the study of local optimal Sobolev inequalities [see for more detail: O. Druet, J. Funct. Anal. 159, 217–242 (1998; Zbl 0923.46035) and E. Hebey, Preprint (2000)]. The theorem is a question of D. L. Johnson and F. Morgan [Indiana Univ. Math. J. 49, 1017–1041 (2000; Zbl 1021.53020)] by modifying a long-standing conjecture with assumption on sectional curvature. In the case, when $$M$$ is compact, the author gets a similar (but stronger) result as a consequence of the first one:
Theorem 2. {Let $$(M, g)$$ be a compact Riemannian manifold of dimension $$n\geq 2$$ with scalar curvature $$S_g<n(n-1)K_0.$$ Then there exists $$V>0$$ such that for any subset $$\Omega$$ of $$M$$ of volume less than or equal to $$V,$$ $| \partial\Omega| _g>| \partial B| _{g_0}$ where $$B$$ is a ball of volume $$| \Omega| _g$$ in the model space $$(M_0, g_0)$$ of constant sectional curvature $$K_0.$$}
Note that this situation was considered first by Johnson and Morgan [loc. cit.] .

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35J20 Variational methods for second-order elliptic equations

### Citations:

Zbl 0923.46035; Zbl 1021.53020
Full Text:

### References:

 [1] Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573 – 598 (French). · Zbl 0371.46011 [2] Thierry Aubin and Yan Yan Li, On the best Sobolev inequality, J. Math. Pures Appl. (9) 78 (1999), no. 4, 353 – 387. · Zbl 0944.46027 [3] Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755 – 782 (1984) (French). · Zbl 0537.53056 [4] Olivier Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 (1998), no. 1, 217 – 242. · Zbl 0923.46035 [5] Olivier Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327 – 346. · Zbl 0934.53028 [6] -, Isoperimetric inequalities on compact manifolds, Geometriae Dedicata (to appear). · Zbl 1025.58014 [7] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001 [8] Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. · Zbl 0981.58006 [9] -, Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Preprint (2000). [10] D. Johnson and F. Morgan, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana University Math. Journal 49, 3 (2000), 1017-1041.CMP 2001:06 · Zbl 1021.53020 [11] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary). [12] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145 – 201. · Zbl 0704.49005 [13] Frank Morgan, Geometric measure theory, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. · Zbl 0819.49024 [14] Michael Struwe, Variational methods, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 34, Springer-Verlag, Berlin, 1996. Applications to nonlinear partial differential equations and Hamiltonian systems. · Zbl 0864.49001 [15] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353 – 372. · Zbl 0353.46018 [16] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. · Zbl 0692.46022
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