2017/01/31 by Isabel M. C. Salavessa
Mathematics · #Curvature #Differential geometry #Divergence (linguistics) #Geometric Analysis and Curvature Flows #Graph #Mean curvature #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #Product (mathematics) #Product topology #Submanifold #math.DG #msc:53C21 #msc:53C24 #msc:53C40 #msc:58C40
paper · pdf · doi:10.1007/s10455-017-9577-x
published as Annals of Global Analysis and Geometry, March 2018, Volume 53, Issue 2, pp 265 - 281 · v4. 20 Pages, we add examples, and correct some constants. v2. 12 pages. We obtain a substancial improvement of the estimation of the mean curvature by using the weighted Cheeger constant recently introduced by Impera at all. in Arxiv 1612.01257
openalex created_date 2017/03/23 · arxiv created 2017/05/27 · openalex publication_date 2017/09/30 · arxiv updated 2018/03/13 · openalex updated_date 2026/08/05
If a graph submanifold (x,f(x)) of a Riemannian warped product space (Mm×eψNn,g=g+e2ψh) is immersed with parallel mean curvature H, then we obtain a Heinz type estimation of the mean curvature. Namely, on each compact domain D of M, m‖H‖≤ (Aψ(∂ D))/(Vψ(D)) holds, where Aψ(∂ D) and Vψ(D) are the ψ-weighted area and volume, respectively. In particular, H=0 if (M,g) has zero weighted Cheeger constant, a concept recently introduced by D. Impera et al. (\cite[Im]). This generalizes the known cases n=1 or ψ=0. We also conclude minimality using a closed calibration, assuming (M,g_*) is complete where g_*=g+e2ψf^*h, and for some constants α≥ δ≥ 0, C1>0 and β∈ [0,1), ‖∇^*ψ‖2g_*≤ δ, Ricciψ,g_*≥ α, and detg(g_*)≤ C1 r2β holds when r→ +∞, where r(x) is the distance function on (M,g_*) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.