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SUMMARY:The Principal Curvature Theorem and its applications to constant m
ean curvature hypersurfaces in Euclidean space
DTSTART;VALUE=DATE-TIME:20190426T120000Z
DTEND;VALUE=DATE-TIME:20190426T140000Z
DTSTAMP;VALUE=DATE-TIME:20220521T224037Z
UID:indico-event-4596@indico.math.cnrs.fr
DESCRIPTION:The so called Principal Curvature Theorem (PCT) is a purely ge
ometric result on the principal curvatures of complete hypersurfaces in Eu
clidean space given by Smyth and Xavier (Invent. Math. 90:443--450\, 1987)
in their proof of Efimov's theorem in dimension greater than two. As anot
her application of the PCT\, they also proved that the only complete hyper
surfaces immersed in \\(\\mathbb{R}^{n+1}\\) with constant mean curvature
\\(H\\neq 0\\) and having non-positive Ricci curvature are the right circu
lar cylinders of the form \\(\\mathbb{R}^{n-1}\\times\\mathbb{S}^1(r)\\)\,
with \\(r>0\\)\, extending to the \\(n\\)-dimensional case a previous res
ult for \\(n=2\\) due to Klotz and Osserman.\n\nIn this lecture we will in
troduce new applications of the PCT to the study of complete hypersurfaces
with constant mean curvature immersed into the Euclidean space \\(\\mathb
b{R}^{n+1}\\)\, and\, more generally\, with constant higher order mean cur
vature. For instance\, among other results\, we will prove that if \\(M^n\
\) is a complete hypersurface in \\(\\mathbb{R}^{n+1}\\) (\\(n\\geq 3\\))
with constant mean curvature \\(H\\neq 0\\) and having two distinct princi
pal curvatures\, one of them being simple\, then \\(\\sup_M\\mathrm{Scal}\
\geq 0\\) and equality holds if and only if \\(M\\) is a right circular cy
linder \\(\\mathbb{R}^{n-1}\\times\\mathbb{S}^1(r)\\)\, with \\(r>0\\).\n\
nOur results in this talk are part of our joint work with S. Carolina Garc
ia-Martinez (Geom. Dedicata 156:31--47\, 2012) and with Josué Meléndez (
Geom. Dedicata 182:117--131\, 2016\; Geom. Dedicata 199:273--280\, 2019).\
n\nhttps://indico.math.cnrs.fr/event/4596/
LOCATION:Tours 1180 (Bât E2)
URL:https://indico.math.cnrs.fr/event/4596/
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