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PDE Real Analysis Seminar
Seminar information archive ～04/23｜Next seminar｜Future seminars 04/24～
| Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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2019/07/23
13:00-14:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Tianling Jin (The Hong Kong University of Science and Technology)
On the isoperimetric ratio over scalar-flat conformal classes (English)
Tianling Jin (The Hong Kong University of Science and Technology)
On the isoperimetric ratio over scalar-flat conformal classes (English)
[ Abstract ]
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality on Euclidean space, and consequently is achieved, if either (i) $n \geq 12$ and the boundary has a nonumbilic point; or (ii) $n \geq 10$, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality on Euclidean space, and consequently is achieved, if either (i) $n \geq 12$ and the boundary has a nonumbilic point; or (ii) $n \geq 10$, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions.
