2019 # Existence of min-max free boundary disks realizing the width of a manifold

**Abstract:** We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lies in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Colding-Minicozzi replacement procedure.

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“…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.…”

confidence: 63%

“…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.…”

confidence: 63%

“…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.…”

confidence: 99%

“…We gives a proof of a Wente type lemma, which is inspired of Theorem A.4 of [24] (see also Theorem 1.100 [35] and proof of Theorem 1.2 [25]).…”

confidence: 99%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%