Tobias Holck Colding

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This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R. We(More)
In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question that Grisha Perelman asked the first author at a dinner in New York City on April 25th of 2003. His question was “what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In(More)
This paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the flat metric).(More)
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key is to understand the structure of an embedded minimal disk in a ball in R. This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see [CM15] for discussion of(More)
Twenty years ago Yau, 56], generalized the classical Liouville theorem of complex analysis to open manifolds with nonnegative Ricci curvature. Speciically, he proved that a positive harmonic function on such a manifold must be constant. This theorem of Yau was considerably generalized by Cheng-Yau (see 15]) by means of a gradient estimate which implies the(More)
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This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale (in fact, Corollary III.3.5 below is used in [CM5]). To describe general planar domains (in(More)