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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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