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A new isoperimetric comparison theorem for surfaces of variable curvature. (English) Zbl 0886.53031

The isoperimetric profile of a Riemannian surface \(M\) is the function \(I_M (v):=\) infimum over the areas of the boundaries of compact subdomains with volume equal to \(v\). The main theorem of this paper is: Let \(M_1\) be the model space with total positive curvature less or equal to 2 such that \(M_1= (\mathbb{R}^2,g_1)\) is a complete, simply connected rotationally symmetric surface of the metric \(g_1= dr^2 +f^2_1 (r)d \theta^2\) with decreasing sectional curvature \(K_{M_1} (r)\). If \(M_2\) is a complete, simply connected Riemannian surface having smaller curvature than \(M_1\) over domains of area \(A\) (i.e., there does not exist a domain in \(M_2\) with area \(A\) such that its total curvature is greater than the total curvature of each domain in \(M_1\) with volume \(A)\) for all \(A\), then \(I_{M_1} (v)\leq I_{M_2} (v)\) for all \(v\). Further, if \(D_{r(v)} \subset M_1\) is the geodesic disk, centered at the origin, of area \(v\), then \(L(\partial D_{r(v)}) =I_{M_1}(v)\) where \(L(\partial D_{r(v)})\) denotes the length of \(\partial D_{r(v)}\).
The proof uses curve shortening. An application gives sharp estimates of the first eigenvalue of the Laplacian. Further, a similar comparison theorem for isoperimetric profiles on compact convex surfaces is derived.

MSC:

53C20 Global Riemannian geometry, including pinching
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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