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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:
<blockquote>Consider [[compact manifold|closed]] [[minimal manifold|minimal]] [[submanifold]]s <math>M^n</math> immersed in the unit sphere <math>S^{n+m}</math> with [[second fundamental form]] of constant length whose square is denoted by <math>\sigma</math>. Is the set of values for <math>\sigma</math> discrete? What is the infimum of these values of <math>\sigma > \frac{n}{2-\frac{1}{m}}</math>?</blockquote>
The first question, i.e., whether the set of values for ''σ'' is discrete, can be reformulated as follows:
<blockquote>Let <math>M^n</math> be a closed minimal submanifold in <math>\mathbb{S}^{n+m}</math> with the second fundamental form of constant length, denote by <math>\mathcal{A}_n</math> the set of all the possible values for the squared length of the second fundamental form of <math>M^n</math>, is <math>\mathcal{A}_n</math> a discrete?</blockquote>
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):
<blockquote>Consider the set of all compact minimal [[hypersurface]]s in <math>S^N</math> with constant scalar curvature. Think of the scalar curvature as a function on this set. Is the [[image (mathematics)|image]] of this function a [[discrete set]] of positive numbers?</blockquote>
Formulated alternatively:
<blockquote>Consider closed minimal hypersurfaces <math>M \subset \mathbb{S}^{n+1}</math> with constant scalar curvature <math>k</math>. Then for each <math>n</math> the set of all possible values for <math>k</math> (or equivalently <math>S</math>) is discrete</blockquote>
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'''):
<blockquote>Let <math>M^n</math> be a closed, minimally immersed hypersurface of the unit sphere <math>S^{n+1}</math> with constant scalar curvature. Then <math>M</math> is isoparametric</blockquote>
Here, <math>S^{n+1}</math> refers to the (n+1)-dimensional sphere, and n ≥ 2.
In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with <math>\sigma + \lambda_2</math> taken instead of <math>\sigma</math>:
<blockquote>Let <math>M^n</math> be a closed, minimally immersed submanifold in the unit sphere <math>\mathbb{S}^{n+m} </math> with constant <math>\sigma + \lambda_2</math>. If <math>\sigma + \lambda_2 > n</math>, then there is a constant <math>\epsilon(n, m) > 0</math> such that<math>\sigma + \lambda_2 > n + \epsilon(n, m)</math></blockquote>
Here, <math>M^n</math> denotes an n-dimensional minimal submanifold; <math>\lambda_2</math> denotes the second largest [[eigenvalue]] of the semi-positive symmetric matrix <math>S := (\left \langle A^\alpha, B^\beta \right \rangle)</math> where <math>A^\alpha</math>s (<math>\alpha = 1, \cdots, m</math>) are the [[shape operator]]s of <math>M</math> with respect to a given (local) normal orthonormal frame. <math>\sigma</math> is rewritable as <math>{\left \Vert \sigma \right \Vert}^2</math>.
Another related conjecture was proposed by [[Robert Bryant (mathematician)]]:
<blockquote>A piece of a minimal hypersphere of <math>\mathbb{S}^4</math> with constant scalar curvature is isoparametric of type <math>g \le 3</math></blockquote>
Formulated alternatively:
<blockquote>Let <math>M \subset \mathbb{S}^4</math> be a minimal hypersurface with constant scalar curvature. Then <math>M</math> is isoparametric</blockquote>
==Chern's conjectures hierarchically==
Put hierarchically and formulated in a single style, Chern's conjectures (without conjectures of Lu and Bryant) can look like this:
* The first version (minimal hypersurfaces conjecture):
<blockquote>Let <math>M</math> be a compact minimal hypersurface in the unit sphere <math>\mathbb{S}^{n+1}</math>. If <math>M</math> has constant scalar curvature, then the possible values of the scalar curvature of <math>M</math> form a discrete set</blockquote>
* The refined/stronger version (isoparametric hypersurfaces conjecture) of the conjecture is the same, but with the "if" part being replaced with this:
<blockquote>If <math>M</math> has constant scalar curvature, then <math>M</math> is isoparametric</blockquote>
* The strongest version replaces the "if" part with:
<blockquote>Denote by <math>S</math> the squared length of the second fundamental form of <math>M</math>. Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>, for <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 5 \}</math>. Then we have:
* For any fixed <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 4 \}</math>, if <math>a_k \le S \le a_{k+1}</math>, then <math>M</math> is isoparametric, and <math>S \equiv a_k</math> or <math>S \equiv a_{k+1}</math>
* If <math>S \ge a_5</math>, then <math>M</math> is isoparametric, and <math>S \equiv a_5</math></blockquote>
Or alternatively:
<blockquote>Denote by <math>A</math> the squared length of the second fundamental form of <math>M</math>. Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>, for <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 5 \}</math>. Then we have:
* For any fixed <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 4 \}</math>, if <math>a_k \le {\left \vert A \right \vert}^2 \le a_{k+1}</math>, then <math>M</math> is isoparametric, and <math>{\left \vert A \right \vert}^2 \equiv a_k</math> or <math>{\left \vert A \right \vert}^2 \equiv a_{k+1}</math>
* If <math>{\left \vert A \right \vert}^2 \ge a_5</math>, then <math>M</math> is isoparametric, and <math>{\left \vert A \right \vert}^2 \equiv a_5</math></blockquote>
One should pay attention to the so-called first and second pinching problems as special parts for Chern.
==Other related and still open problems==
Besides the conjectures of Lu and Bryant, there're also others:
In 1983, Chia-Kuei Peng and [[Chuu-Lian Terng]] proposed the problem related to Chern:
<blockquote>Let <math>M</math> be a <math>n</math>-dimensional closed minimal hypersurface in <math>S^{n+1}, n \ge 6</math>. Does there exist a positive constant <math>\delta(n)</math> depending only on <math>n</math> such that if <math>n \le n + \delta(n)</math>, then <math>S \equiv n</math>, i.e., <math>M</math> is one of the [[Clifford torus]] <math>S^k(\sqrt{\frac{k}{n}}) \times S^{n-k}(\sqrt{\frac{n-k}{n}}), k = 1, 2, ..., n-1</math>?</blockquote>
In 2017, Li Lei, Hongwei Xu and Zhiyuan Xu proposed 2 Chern-related problems.
The 1st one was inspired by [[Yau's conjecture on the first eigenvalue]]:
<blockquote>Let <math>M</math> be an <math>n</math>-dimensional compact minimal hypersurface in <math>\mathbb{S}^{n+1}</math>. Denote by <math>\lambda_1(M)</math> the first [[eigenvalue]] of the [[Laplace operator]] acting on functions over <math>M</math>:
* Is it possible to prove that if <math>M</math> has constant scalar curvature, then <math>\lambda_1(M) = n</math>?
* Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>. Is it possible to prove that if <math>a_k \le S \le a_{k+1}</math> for some <math>k \in \{ m \in \mathbb{Z}^+ ; 2 \le m \le 4 \}</math>, or <math>S \ge a_5</math>, then <math>\lambda_1(M) = n</math>?</blockquote>
The second is their own '''generalized Chern's conjecture for hypersurfaces with constant mean curvature''':
<blockquote>Let <math>M</math> be a closed hypersurface with constant mean curvature <math>H</math> in the unit sphere <math>\mathbb{S}^{n+1}</math>:
* Assume that <math>a \le S \le b</math>, where <math>a < b</math> and <math>\left [ a, b \right ] \cap I = \left \lbrace a, b \right \rbrace</math>. Is it possible to prove that <math>S \equiv a</math> or <math>S \equiv b</math>, and <math>M</math> is an isoparametric hypersurface in <math>\mathbb{S}^{n+1}</math>?
* Suppose that <math>S \le c</math>, where <math>c = \sup_{t \in I}{t}</math>. Can one show that <math>S \equiv c</math>, and <math>M</math> is an isoparametric hypersurface in <math>\mathbb{S}^{n+1}</math>?</blockquote>
==Further reading==
* S.S. Chern, Minimal Submanifolds in a Riemannian Manifold, ([[mimeograph]]ed in 1968), Department of Mathematics Technical Report 19 (New Series), [[University of Kansas]], 1968
* S.S. Chern, Brief survey of minimal submanifolds, Differentialgeometrie im Großen, volume 4 (1971), [[Mathematisches Forschungsinstitut Oberwolfach]], pp. 43–60
* S.S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields: Proceedings of a Conference in honor of Professor [[Marshall Stone]], held at the [[University of Chicago]], May 1968 (1970), [[Springer-Verlag]], pp. 59-75
* S.T. Yau, Seminar on Differential Geometry (Annals of Mathematics Studies, Volume 102), [[Princeton University Press]] (1982), pp. 669–706, problem 105
* L. Verstraelen, Sectional curvature of minimal submanifolds, Proceedings of the Workshop on Differential Geometry (1986), [[University of Southampton]], pp. 48–62
* M. Scherfner and S. Weiß, Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres, Süddeutsches Kolloquium über Differentialgeometrie, volume 33 (2008), Institut für Diskrete Mathematik und Geometrie, [[Technische Universität Wien]], pp. 1–13
* Z. Lu, Normal scalar curvature conjecture and its applications, Journal of Functional Analysis, volume 261 (2011), pp. 1284–1308
* Liquid error: wrong number of arguments (1 for 2)
* C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
* Liquid error: wrong number of arguments (1 for 2)
[[Category:Conjectures]]
[[Category:Differential geometry]]
'''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:
<blockquote>Consider [[compact manifold|closed]] [[minimal manifold|minimal]] [[submanifold]]s <math>M^n</math> immersed in the unit sphere <math>S^{n+m}</math> with [[second fundamental form]] of constant length whose square is denoted by <math>\sigma</math>. Is the set of values for <math>\sigma</math> discrete? What is the infimum of these values of <math>\sigma > \frac{n}{2-\frac{1}{m}}</math>?</blockquote>
The first question, i.e., whether the set of values for ''σ'' is discrete, can be reformulated as follows:
<blockquote>Let <math>M^n</math> be a closed minimal submanifold in <math>\mathbb{S}^{n+m}</math> with the second fundamental form of constant length, denote by <math>\mathcal{A}_n</math> the set of all the possible values for the squared length of the second fundamental form of <math>M^n</math>, is <math>\mathcal{A}_n</math> a discrete?</blockquote>
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):
<blockquote>Consider the set of all compact minimal [[hypersurface]]s in <math>S^N</math> with constant scalar curvature. Think of the scalar curvature as a function on this set. Is the [[image (mathematics)|image]] of this function a [[discrete set]] of positive numbers?</blockquote>
Formulated alternatively:
<blockquote>Consider closed minimal hypersurfaces <math>M \subset \mathbb{S}^{n+1}</math> with constant scalar curvature <math>k</math>. Then for each <math>n</math> the set of all possible values for <math>k</math> (or equivalently <math>S</math>) is discrete</blockquote>
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'''):
<blockquote>Let <math>M^n</math> be a closed, minimally immersed hypersurface of the unit sphere <math>S^{n+1}</math> with constant scalar curvature. Then <math>M</math> is isoparametric</blockquote>
Here, <math>S^{n+1}</math> refers to the (n+1)-dimensional sphere, and n ≥ 2.
In 2008, Zhiqin Lu proposed a conjecture similar to that of Chern, but with <math>\sigma + \lambda_2</math> taken instead of <math>\sigma</math>:
<blockquote>Let <math>M^n</math> be a closed, minimally immersed submanifold in the unit sphere <math>\mathbb{S}^{n+m} </math> with constant <math>\sigma + \lambda_2</math>. If <math>\sigma + \lambda_2 > n</math>, then there is a constant <math>\epsilon(n, m) > 0</math> such that<math>\sigma + \lambda_2 > n + \epsilon(n, m)</math></blockquote>
Here, <math>M^n</math> denotes an n-dimensional minimal submanifold; <math>\lambda_2</math> denotes the second largest [[eigenvalue]] of the semi-positive symmetric matrix <math>S := (\left \langle A^\alpha, B^\beta \right \rangle)</math> where <math>A^\alpha</math>s (<math>\alpha = 1, \cdots, m</math>) are the [[shape operator]]s of <math>M</math> with respect to a given (local) normal orthonormal frame. <math>\sigma</math> is rewritable as <math>{\left \Vert \sigma \right \Vert}^2</math>.
Another related conjecture was proposed by [[Robert Bryant (mathematician)]]:
<blockquote>A piece of a minimal hypersphere of <math>\mathbb{S}^4</math> with constant scalar curvature is isoparametric of type <math>g \le 3</math></blockquote>
Formulated alternatively:
<blockquote>Let <math>M \subset \mathbb{S}^4</math> be a minimal hypersurface with constant scalar curvature. Then <math>M</math> is isoparametric</blockquote>
==Chern's conjectures hierarchically==
Put hierarchically and formulated in a single style, Chern's conjectures (without conjectures of Lu and Bryant) can look like this:
* The first version (minimal hypersurfaces conjecture):
<blockquote>Let <math>M</math> be a compact minimal hypersurface in the unit sphere <math>\mathbb{S}^{n+1}</math>. If <math>M</math> has constant scalar curvature, then the possible values of the scalar curvature of <math>M</math> form a discrete set</blockquote>
* The refined/stronger version (isoparametric hypersurfaces conjecture) of the conjecture is the same, but with the "if" part being replaced with this:
<blockquote>If <math>M</math> has constant scalar curvature, then <math>M</math> is isoparametric</blockquote>
* The strongest version replaces the "if" part with:
<blockquote>Denote by <math>S</math> the squared length of the second fundamental form of <math>M</math>. Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>, for <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 5 \}</math>. Then we have:
* For any fixed <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 4 \}</math>, if <math>a_k \le S \le a_{k+1}</math>, then <math>M</math> is isoparametric, and <math>S \equiv a_k</math> or <math>S \equiv a_{k+1}</math>
* If <math>S \ge a_5</math>, then <math>M</math> is isoparametric, and <math>S \equiv a_5</math></blockquote>
Or alternatively:
<blockquote>Denote by <math>A</math> the squared length of the second fundamental form of <math>M</math>. Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>, for <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 5 \}</math>. Then we have:
* For any fixed <math>k \in \{ m \in \mathbb{Z}^+ ; 1 \le m \le 4 \}</math>, if <math>a_k \le {\left \vert A \right \vert}^2 \le a_{k+1}</math>, then <math>M</math> is isoparametric, and <math>{\left \vert A \right \vert}^2 \equiv a_k</math> or <math>{\left \vert A \right \vert}^2 \equiv a_{k+1}</math>
* If <math>{\left \vert A \right \vert}^2 \ge a_5</math>, then <math>M</math> is isoparametric, and <math>{\left \vert A \right \vert}^2 \equiv a_5</math></blockquote>
One should pay attention to the so-called first and second pinching problems as special parts for Chern.
==Other related and still open problems==
Besides the conjectures of Lu and Bryant, there're also others:
In 1983, Chia-Kuei Peng and [[Chuu-Lian Terng]] proposed the problem related to Chern:
<blockquote>Let <math>M</math> be a <math>n</math>-dimensional closed minimal hypersurface in <math>S^{n+1}, n \ge 6</math>. Does there exist a positive constant <math>\delta(n)</math> depending only on <math>n</math> such that if <math>n \le n + \delta(n)</math>, then <math>S \equiv n</math>, i.e., <math>M</math> is one of the [[Clifford torus]] <math>S^k(\sqrt{\frac{k}{n}}) \times S^{n-k}(\sqrt{\frac{n-k}{n}}), k = 1, 2, ..., n-1</math>?</blockquote>
In 2017, Li Lei, Hongwei Xu and Zhiyuan Xu proposed 2 Chern-related problems.
The 1st one was inspired by [[Yau's conjecture on the first eigenvalue]]:
<blockquote>Let <math>M</math> be an <math>n</math>-dimensional compact minimal hypersurface in <math>\mathbb{S}^{n+1}</math>. Denote by <math>\lambda_1(M)</math> the first [[eigenvalue]] of the [[Laplace operator]] acting on functions over <math>M</math>:
* Is it possible to prove that if <math>M</math> has constant scalar curvature, then <math>\lambda_1(M) = n</math>?
* Set <math>a_k = (k - \operatorname{sgn}(5-k))n</math>. Is it possible to prove that if <math>a_k \le S \le a_{k+1}</math> for some <math>k \in \{ m \in \mathbb{Z}^+ ; 2 \le m \le 4 \}</math>, or <math>S \ge a_5</math>, then <math>\lambda_1(M) = n</math>?</blockquote>
The second is their own '''generalized Chern's conjecture for hypersurfaces with constant mean curvature''':
<blockquote>Let <math>M</math> be a closed hypersurface with constant mean curvature <math>H</math> in the unit sphere <math>\mathbb{S}^{n+1}</math>:
* Assume that <math>a \le S \le b</math>, where <math>a < b</math> and <math>\left [ a, b \right ] \cap I = \left \lbrace a, b \right \rbrace</math>. Is it possible to prove that <math>S \equiv a</math> or <math>S \equiv b</math>, and <math>M</math> is an isoparametric hypersurface in <math>\mathbb{S}^{n+1}</math>?
* Suppose that <math>S \le c</math>, where <math>c = \sup_{t \in I}{t}</math>. Can one show that <math>S \equiv c</math>, and <math>M</math> is an isoparametric hypersurface in <math>\mathbb{S}^{n+1}</math>?</blockquote>
==Further reading==
* S.S. Chern, Minimal Submanifolds in a Riemannian Manifold, ([[mimeograph]]ed in 1968), Department of Mathematics Technical Report 19 (New Series), [[University of Kansas]], 1968
* S.S. Chern, Brief survey of minimal submanifolds, Differentialgeometrie im Großen, volume 4 (1971), [[Mathematisches Forschungsinstitut Oberwolfach]], pp. 43–60
* S.S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields: Proceedings of a Conference in honor of Professor [[Marshall Stone]], held at the [[University of Chicago]], May 1968 (1970), [[Springer-Verlag]], pp. 59-75
* S.T. Yau, Seminar on Differential Geometry (Annals of Mathematics Studies, Volume 102), [[Princeton University Press]] (1982), pp. 669–706, problem 105
* L. Verstraelen, Sectional curvature of minimal submanifolds, Proceedings of the Workshop on Differential Geometry (1986), [[University of Southampton]], pp. 48–62
* M. Scherfner and S. Weiß, Towards a proof of the Chern conjecture for isoparametric hypersurfaces in spheres, Süddeutsches Kolloquium über Differentialgeometrie, volume 33 (2008), Institut für Diskrete Mathematik und Geometrie, [[Technische Universität Wien]], pp. 1–13
* Z. Lu, Normal scalar curvature conjecture and its applications, Journal of Functional Analysis, volume 261 (2011), pp. 1284–1308
* Liquid error: wrong number of arguments (1 for 2)
* C.K. Peng, C.L. Terng, Minimal hypersurfaces of sphere with constant scalar curvature, Annals of Mathematics Studies, volume 103 (1983), pp. 177–198
* Liquid error: wrong number of arguments (1 for 2)
[[Category:Conjectures]]
[[Category:Differential geometry]]
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