2022 Preprint# Min-max free boundary minimal surface with genus at least one

**Abstract:** In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus g ≥ 1 and m ≥ 1 ideal boundary components. We show that the width for the area functional can be achieved by a bubble tree limit consisting of branched genus g free boundary minimal surfaces with nodes, and possibly finitely many branched minimal spheres and free boundary minimal disks. Our result extends the min-max theory by [4][10][9] to all genera.

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“…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%

“…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%