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Title:
One-cycle sweepout estimates of essential surfaces in closed Riemannian manifolds
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Abstract:
Abstract: We present new-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose one-cycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption.
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