BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Costante Bellettini (UCL) DTSTART:20210525T124500Z DTEND:20210525T134500Z DTSTAMP:20260423T040151Z UID:BOWL/24 DESCRIPTION:Title: Ex istence of hypersurfaces with prescribed mean-curvature\nby Costante B ellettini (UCL) as part of B.O.W.L Geometry Seminar\n\n\nAbstract\nLet N b e a compact Riemannian manifold of dimension 3 or higher\, and g a Lipschi tz non-negative (or non-positive) function on N. We prove that there exist s a closed hypersurface M whose mean curvature attains the values prescrib ed by g (joint work with Neshan Wickramasekera\, Cambridge). Except possib ly for a small singular set (of codimension 7 or higher)\, the hypersurfac e M is C^2 immersed and two-sided (it admits a global unit normal)\; the s calar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely\, the immersion is a quasi-embedding\, namely the o nly non-embedded points are caused by tangential self-intersections: aroun d such a non-embedded point\, the local structure is given by two disks\, lying on one side of each other\, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (p rescribed-mean-curvature) hypersurfaces is obtained when g is a constant\, in which the above result gives a CMC (constant-mean-curvature) hypersurf ace for any prescribed value of the mean curvature. The construction of M is carried out largely by means of PDE principles: (i) a minmax for an All en–Cahn (or Modica-Mortola) energy\, involving a parameter that\, when s ent to 0\, leads to an interface from which the desired PMC hypersurface i s extracted\; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments\, to obtain regularity conclusions for said interface\; (iii) pa rabolic semi-linear PDE (together with specific features of the Allen-Cahn framework)\, to tackle cancellation phenomena that can happen when sendin g to 0 the Allen-Cahn parameter.\n LOCATION:https://researchseminars.org/talk/BOWL/24/ END:VEVENT END:VCALENDAR