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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:20221007T193800Z
UID:indico-event-4596@indico.math.cnrs.fr
DESCRIPTION:Speakers: Luis Alias (Universté de Murcia)\n\nThe so called P
rincipal Curvature Theorem (PCT) is a purely geometric result on the princ
ipal curvatures of complete hypersurfaces in Euclidean space given by Smyt
h and Xavier (Invent. Math. 90:443--450\, 1987) in their proof of Efimov's
theorem in dimension greater than two. As another application of the PCT\
, they also proved that the only complete hypersurfaces immersed in \\(\\m
athbb{R}^{n+1}\\) with constant mean curvature \\(H\\neq 0\\) and having n
on-positive Ricci curvature are the right circular cylinders of the form \
\(\\mathbb{R}^{n-1}\\times\\mathbb{S}^1(r)\\)\, with \\(r>0\\)\, extending
to the \\(n\\)-dimensional case a previous result for \\(n=2\\) due to Kl
otz and Osserman.\n\nIn this lecture we will introduce new applications of
the PCT to the study of complete hypersurfaces with constant mean curvatu
re immersed into the Euclidean space \\(\\mathbb{R}^{n+1}\\)\, and\, more
generally\, with constant higher order mean curvature. For instance\, amon
g other results\, we will prove that if \\(M^n\\) is a complete hypersurfa
ce in \\(\\mathbb{R}^{n+1}\\) (\\(n\\geq 3\\)) with constant mean curvatur
e \\(H\\neq 0\\) and having two distinct principal curvatures\, one of the
m being simple\, then \\(\\sup_M\\mathrm{Scal}\\geq 0\\) and equality hold
s if and only if \\(M\\) is a right circular cylinder \\(\\mathbb{R}^{n-1}
\\times\\mathbb{S}^1(r)\\)\, with \\(r>0\\).\n\nOur results in this talk a
re part of our joint work with S. Carolina Garcia-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:1180 (Bât E2) (Tours)
URL:https://indico.math.cnrs.fr/event/4596/
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