Displaying similar documents to “A genericity theorem for algebraic stacks and essential dimension of hypersurfaces”

A characterization of a certain real hypersurface of type ( A 2 ) in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda (2017)

Czechoslovak Mathematical Journal

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In the class of real hypersurfaces M 2 n - 1 isometrically immersed into a nonflat complex space form M ˜ n ( c ) of constant holomorphic sectional curvature c ( 0 ) which is either a complex projective space P n ( c ) or a complex hyperbolic space H n ( c ) according as c > 0 or c < 0 , there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

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We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n 3 ) with constant ( r + 1 ) -th mean curvature H r + 1 immersed into the Euclidean sphere 𝕊 n + 1 , where r { 1 , ... , n - 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of 𝕊 n + 1 , a region that is determined by a totally umbilical sphere of 𝕊 n + 1 . We also provide a rigidity result for such hypersurfaces.

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a domain Ω of m + 1 and a k -dimensional non-degenerate minimal submanifold K of Ω with 1 k m - 1 , we prove the existence of a family of embedded constant mean curvature hypersurfaces in Ω which as their mean curvature tends to infinity concentrate along K and intersecting Ω perpendicularly along their boundaries.

Pointed k -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

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Let S be a Riemann surface. Let 3 be the 3 -dimensional hyperbolic space and let 3 be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping ϕ : S 3 = ^ . If i : S 3 is a convex immersion, and if N is its exterior normal vector field, we define the Gauss lifting, ı ^ , of i by ı ^ = N . Let n : U 3 3 be the Gauss-Minkowski mapping. A solution to the Plateau problem ( S , ϕ ) is a convex immersion i of constant Gaussian curvature equal to k ( 0 , 1 ) such that the Gauss lifting ( S , ı ^ ) is complete and n ı ^ = ϕ . In this...

J -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

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We give in 6 a real analytic almost complex structure J , a real analytic hypersurface M and a vector v in the Levi null set at 0 of M , such that there is no germ of J -holomorphic disc γ included in M with γ ( 0 ) = 0 and γ x ( 0 ) = v , although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J , we give sufficient conditions under which there exists such a germ of disc.

Monodromy of a family of hypersurfaces

Vincenzo Di Gennaro, Davide Franco (2009)

Annales scientifiques de l'École Normale Supérieure

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Let Y be an ( m + 1 ) -dimensional irreducible smooth complex projective variety embedded in a projective space. Let Z be a closed subscheme of Y , and δ be a positive integer such that Z , Y ( δ ) is generated by global sections. Fix an integer d δ + 1 , and assume the general divisor X | H 0 ( Y , Z , Y ( d ) ) | is smooth. Denote by H m ( X ; ) Z van the quotient of H m ( X ; ) by the cohomology of Y and also by the cycle classes of the irreducible components of dimension m of Z . In the present paper we prove that the monodromy representation on H m ( X ; ) Z van for the family...

A short note on f -biharmonic hypersurfaces

Selcen Y. Perktaş, Bilal E. Acet, Adara M. Blaga (2020)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper we give some properties of f -biharmonic hypersurfaces in real space forms. By using the f -biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the f -biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider f -biharmonic vertical cylinders in S 2 × .

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p 1/p , aj ∈ ℂ , such that ( x - 1 ) k divides P(x). For n ∈ ℕ and L > 0 let κ ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L m a x 1 j n | a j | , a j , such that ( x - 1 ) k divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that c 1 ( n / L ) - 1 κ ( n , L ) c 2 ( n / L ) for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Hardness of embedding simplicial complexes in d

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let 𝙴𝙼𝙱𝙴𝙳 k d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k , does there exist a (piecewise linear) embedding of K into d ? Known results easily imply polynomiality of 𝙴𝙼𝙱𝙴𝙳 k 2 ( k = 1 , 2 ; the case k = 1 , d = 2 is graph planarity) and of 𝙴𝙼𝙱𝙴𝙳 k 2 k for all k 3 . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that 𝙴𝙼𝙱𝙴𝙳 d d and 𝙴𝙼𝙱𝙴𝙳 ( d - 1 ) d are undecidable for each d 5 . Our main result is NP-hardness of 𝙴𝙼𝙱𝙴𝙳 2 4 and, more generally, of 𝙴𝙼𝙱𝙴𝙳 k d for all...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

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Let V be the complex vector space of homogeneous linear polynomials in the variables x 1 , ... , x m . Suppose G is a subgroup of S m , and χ is an irreducible character of G . Let H d ( G , χ ) be the symmetry class of polynomials of degree d with respect to G and χ . For any linear operator T acting on V , there is a (unique) induced operator K χ ( T ) End ( H d ( G , χ ) ) acting on symmetrized decomposable polynomials by K χ ( T ) ( f 1 * f 2 * ... * f d ) = T f 1 * T f 2 * ... * T f d . In this paper, we show that the representation T K χ ( T ) of the general linear group G L ( V ) is equivalent to the direct sum of χ ( 1 ) copies...