Curve shortening flow and the Banchoff-Pohl inequality on surfaces of nonpositive curvature (Q1591728)

This paper is part of the author's doctoral thesis [\textit{B. Süssmann}, Stuttgart: Univ. Stuttgart, Mathematisches Institut B, 93 S. (1998; Zbl 0909.53042)] and presents the following Banchoff-Pohl inequality for nonsimple closed curves \(C\) on smooth, simply connected, complete Riemann surfaces \(M\) of nonpositive Gauss curvature \(K\): \(L^2\geq 4\pi\int_Mw^2(x) dA-K_0(\int_M|w(x)|dA)^2\), where \(L\) is the length of the curve, \(w(x)\) the winding number of \(x\in M/C\) w.r.t. \(C\), \(dA\) the area element of the Riemann surface and \(K_0=\max_{x\in \text{conv}(C)}K(x)\) with \(\text{conv}(C)\) the convex hull of \(C\). The proof uses the curve shortening flow on surfaces. This result has been obtained independently by Howard [\textit{R. Howard}, Proc. Am. Math. Soc. 126, No. 9, 2779-2787 (1998; Zbl 0902.53048)].