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- Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consider closed submanifolds immersed in the unit sphere with second fundamental form of constant length whose square is denoted by . Is the set of values for discrete? What is the infimum of these values of ? The first question, i.e., whether the set of values for σ is discrete, can be reformulated as follows: Let be a closed minimal submanifold in with the second fundamental form of constant length, denote by the set of all the possible values for the squared length of the second fundamental form of , is a discrete? Its affirmative hand, more general than the Chern's conjecture for hypersurfaces, sometimes also referred to as the Chern's conjecture and is still, as of 2018, unanswered even with M as a hypersurface (Chern proposed this special case to the Shing-Tung Yau's open problems' list in differential geometry in 1982): Consider the set of all compact minimal hypersurfaces in with constant scalar curvature. Think of the scalar curvature as a function on this set. Is the image of this function a discrete set of positive numbers? Formulated alternatively: Consider closed minimal hypersurfaces with constant scalar curvature . Then for each the set of all possible values for (or equivalently ) is discrete This became known as the Chern's conjecture for minimal hypersurfaces in spheres (or Chern's conjecture for minimal hypersurfaces in a sphere) This hypersurface case was later, thanks to progress in isoparametric hypersurfaces' studies, given a new formulation, now known as Chern's conjecture for isoparametric hypersurfaces in spheres (or Chern's conjecture for isoparametric hypersurfaces in a sphere): Let be a closed, minimally immersed hypersurface of the unit sphere with constant scalar curvature. Then is isoparametric Here, refers to the (n+1)-dimensional sphere, and n ≥ 2. In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with taken instead of : Let be a closed, minimally immersed submanifold in the unit sphere with constant . If , then there is a constant such that Here, denotes an n-dimensional minimal submanifold; denotes the second largest eigenvalue of the semi-positive symmetric matrix where s are the shape operators of with respect to a given (local) normal orthonormal frame. is rewritable as . Another related conjecture was proposed by Robert Bryant (mathematician): A piece of a minimal hypersphere of with constant scalar curvature is isoparametric of type Formulated alternatively: Let be a minimal hypersurface with constant scalar curvature. Then is isoparametric (en)
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| rdfs:comment
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- Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consider closed submanifolds immersed in the unit sphere with second fundamental form of constant length whose square is denoted by . Is the set of values for discrete? What is the infimum of these values of ? The first question, i.e., whether the set of values for σ is discrete, can be reformulated as follows: Formulated alternatively: Here, refers to the (n+1)-dimensional sphere, and n ≥ 2. (en)
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