Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications

Abstract

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity.One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N OE0; T , where N is a closed manifold with a certain spectral gap. Examples of such N 's are all (round) spheres S n for n 1 and all Zoll surfaces.Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.

Width and finite extinction time of Ricci flow

“…In [8] we looked at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold (see Figure 2) and for 3-manifolds without aspherical summands (like a homotopy 3-sphere) when the metric evolve by the Ricci flow, the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time.…”

“…When M is prime and nonaspherical, then it follows by standard topology that 3 .M / is nontrivial (see, eg, Colding and Minicozzi [8]). For such an M , fix a nontrivial homotopy classˇ2 .…”

“…This is an expository article of our paper [8] with complete proofs intended for a general nonspecialist audience. The results are two-fold.…”

“…in [25] and by Colding and Minicozzi in [8]; see Perelman [25] also for applications to the elliptic part of geometrization. A conformal map to a long thin surface with small area has little energy.…”

The classical theory of minimal surfaces

“…Therefore, to integrate S M holomorphically one needs to find a holomorphic curve With all these ingredients, one needs to check that for every minimal surface M ∈ M, the function u = u(z) defined by equation (15) [23]). The aforementioned control on the Laurent expansions in poles of u t , coming from the integration of the Cauchy problem for the KdV equation, is enough to prove that the corresponding meromorphic function g t associated to u t by (15) has the correct behavior in poles and zeros; this property, together with the fact that both S M , S * M preserve infinitesimally the complex periods along any closed curve in C/ i , suffices to show that the Weierstrass data (g t , dz) solves the period problem and defines M t ∈ M with the desired properties.…”