Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature
In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We inves...
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doaj-8ec2c6b25ba1427bb9e79035f264b8b72020-11-25T03:10:25ZengWorld Scientific PublishingBulletin of Mathematical Sciences1664-36071664-36152020-04-011012050008-12050008-2410.1142/S166436072050008310.1142/S1664360720500083Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvatureMario Santilli0Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, GermanyIn this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.http://www.worldscientific.com/doi/pdf/10.1142/S1664360720500083area blow-up setvarifoldsgeneralized second fundamental form, generalized gauss mapalmgren’s geometric inequalitysoap bubbles |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Mario Santilli |
spellingShingle |
Mario Santilli Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature Bulletin of Mathematical Sciences area blow-up set varifolds generalized second fundamental form, generalized gauss map almgren’s geometric inequality soap bubbles |
author_facet |
Mario Santilli |
author_sort |
Mario Santilli |
title |
Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature |
title_short |
Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature |
title_full |
Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature |
title_fullStr |
Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature |
title_full_unstemmed |
Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature |
title_sort |
normal bundle and almgren’s geometric inequality for singular varieties of bounded mean curvature |
publisher |
World Scientific Publishing |
series |
Bulletin of Mathematical Sciences |
issn |
1664-3607 1664-3615 |
publishDate |
2020-04-01 |
description |
In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled. |
topic |
area blow-up set varifolds generalized second fundamental form, generalized gauss map almgren’s geometric inequality soap bubbles |
url |
http://www.worldscientific.com/doi/pdf/10.1142/S1664360720500083 |
work_keys_str_mv |
AT mariosantilli normalbundleandalmgrensgeometricinequalityforsingularvarietiesofboundedmeancurvature |
_version_ |
1724658750819663872 |