Conformal curvature flows on compact manifold of negative Yamabe constant

Abstract

We study some conformal curvature flows related to prescribed curvature problems on a smooth compact Riemannian manifold (M, g 0) with or without boundary, which is of negative (generalized) Yamabe constant, including scalar curvature flow and conformal mean curvature flow. Using such flows, we show that there exists a unique conformal metric of g 0 such that its scalar curvature in the interior or mean curvature curvature on the boundary is equal to any prescribed negative smooth function, which partially recovers the results of Kazdan-Warner, Aubin and Escobar. We also study the soliton to some Yamabe-type flow on a compact manifold with smooth boundary.

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

“…The above theorem extends the first and second authors' results in [12]. We point out that it follows from Theorem 1.2 that there exists a conformal metric of g 0 such that its scalar curvature equals f and its mean curvature curvature equals c ∞ h for some positive constant c ∞ = c ∞ (a, b, f, h).…”

Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry

“…The problem of prescribing the scalar curvature or the mean curvature has been studied on manifolds with boundary. See [12,16,20,21,25,39,46] for example. In this paper, we consider the following problem of prescribing the mean curvature on the unit ball, which is a natural analogy of the prescribing scalar curvature problem: Let B n+1 be the (n + 1)-dimensional unit ball equipped with the flat metric g 0 , that is,…”

“…This proves(3.11). It follows from (3.11) that1 f (S) n−1/n + < 1 f (x c ) n−1/n for x c ∈ F with Δ S n f (x c ) 0,(3 12).…”

The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds

“…This problem was initially studied by J. Escobar [23] in the case of 3 ≤ n ≤ 5 or n ≥ 6 and the boundary has a non-umbilic point, later by S. Brendle-S. Chen [12] in the case of n ≥ 6 and the boundary is umbilic, assuming the validity of the Positive Mass Theorem (PMT). For recent associated curvature flows, readers are referred to [9,14,5] and the references therein. The second case is concerned with the existence of scalar-flat conformal metrics with constant boundary mean curvature under the condition that the corresponding generalized Yamabe constant is finite.…”

“…More recently, without the PMT, M. Mayer and C. Ndiaye in [31] studied the remaining cases, but in general the solution they obtained is not a minimizer of the associated energy functional. See [9,14,2] etc for the related conformal curvature flows. The third case is concerned with the existence of conformal metrics with (non-zero) constant scalar curvature and (non-zero) constant boundary mean curvature.…”