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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
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