Min-max minimal disks with free boundary in Riemannian manifolds

Abstract

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in R n to the Riemannian setting.More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2-disk in R 2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit 2-disk proved by Colding and Minicozzi.

Min-max free boundary minimal surface with genus at least one

“…In the free boundary case, in order to perform the harmonic replacement procedure, one needs to get the uniqueness of free boundary harmonic map with small energy with respect to the partial Dirichlet boundary data. It's much harder in the general case for free boundary harmonic map and was solved by Zhou, Lin and Ao in [10], and by Laurain and Petrides in [9]. In this paper, we extend the results of [10] and [9] to all genera free boundary surfaces.…”

“…It's much harder in the general case for free boundary harmonic map and was solved by Zhou, Lin and Ao in [10], and by Laurain and Petrides in [9]. In this paper, we extend the results of [10] and [9] to all genera free boundary surfaces. Now we state the main result.…”

“…Generalizations for tori and higher genus surfaces was performed by Zhou in [16] [17]. Following the work by Colding and Minicozzi, the free boundary minimal disk was proved in [10] by Zhou, Lin and Ao, and in [9] by Laurain and Petrides. In the free boundary case, in order to perform the harmonic replacement procedure, one needs to get the uniqueness of free boundary harmonic map with small energy with respect to the partial Dirichlet boundary data.…”

Existence of free boundary disks with constant mean curvature in $\mathbb{R}^3$

“…Both authors used the α-energy introduced by Sacks-Uhlenbeck [34], with the result in [36] later reproduced using a heat equation as a special case of the main theorem in [38]. In turn, [12] was refined by Lin-Sun-Zhou [27], and independently by Laurain-Petrides [21], using the harmonic replacement procedure developed by Colding and Minicozzi [8]. The very recent work of Sun [41] extends [27,21] from disks to surfaces of other topological types.…”

“…In turn, [12] was refined by Lin-Sun-Zhou [27], and independently by Laurain-Petrides [21], using the harmonic replacement procedure developed by Colding and Minicozzi [8]. The very recent work of Sun [41] extends [27,21] from disks to surfaces of other topological types. On the other hand, in a series of papers (for instance [33,32,31]), an alternative mapping approach together with a regularity theory have been developed by Rivière and Pigati in the case of closed surfaces, and then adapted by Pigati [30] to obtain free boundary minimal surfaces.…”