Theory Of Hypersurfaces Research Articles

Hodge Ideals

We use methods from birational geometry to study M. Saito's Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. We analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.

A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n + 1 . The framework is that of the tangent sphere bundle of M . We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler–Lagrange differential systems.

Min–max theory for constant mean curvature hypersurfaces

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover, if $c$ is nonzero then our min-max solution always has multiplicity one.

The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to $$-\varepsilon ^2\Delta u + W'(u)=0$$ on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if $$\varepsilon $$ is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as $$\varepsilon \rightarrow 0$$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $$\varepsilon $$ , as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as $$\varepsilon \rightarrow 0$$ , around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with $$\mathrm{Ric}_M>0$$ , the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.

Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $\rho^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of $\rho^{\bot}$.

Min-max theory for minimal hypersurfaces with boundary

In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these theorems holds also for the case of the free boundary minimal surfaces.

Min–max for phase transitions and the existence of embedded minimal hypersurfaces

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren–Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min–max theories.

Orthogonal multiplications of type $$[3,4,p], p\le 12$$ [ 3 , 4 , p ] , p ≤ 12

We describe the moduli space of orthogonal multiplications of type $$[3,4,p], p\le 12,$$ and its application to the hypersurface theory.

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from [4]. As a consequence of this, we obtain sharp (up to e losses) Strichartz estimates for the hyperbolic Schrodinger equation on the torus. Our second main result is an l 2 decoupling for nondegenerate curves, which has implications for Vinogradov’s mean value theorem.

Self-shrinkers to the mean curvature flow asymptotic to isoparametric cones

In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C C and lying outside C C . We call a cone C C in R n + 1 \mathbb {R}^{n+1} an isoparametric cone if C C is the cone over a compact embedded isoparametric hypersurface Γ ⊂ S n \Gamma \subset \mathbb {S}^n . The theory of isoparametic hypersurfaces is extremely rich, and there are infinitely many distinct classes of examples, each with infinitely many members.

On the syzygies and Hodge theory of nodal hypersurfaces

We give lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in $$\mathbb {P}^3$$ is improved and applications to deformation theory of nodal surfaces are given.

Optimal Bilinear Restriction Estimates for General Hypersurfaces and the Role of the Shape Operator

It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow from the corresponding linear estimates. This subject was extensively studied for conic and parabolic surfaces with sharp results proved by Wolff and Tao, and with later generalizations by Lee. In this paper we provide a unified theory for general hypersurfaces and clarify the role of curvature in this problem, by making statements in terms of the shape operators of the hypersurfaces involved.

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures. Here a k-th order mean curvature Qkν (k ≥ 1) of a submanifold Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the shape operator with respect to the unit normal vector ν. We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures, then the submanifold M itself has some constant higher order mean curvatures Qkν independent of the choice of ν. Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained. In particular, we generalize several classical results in isoparametric theory given by E. Cartan, K. Nomizu, H. F. M¨unzner, Q. M. Wang, et al. As an application, we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.

Min-max theory for minimal hypersurfaces with boundary

In this thesis we present a result concerning existence and regularity of minimal surfaces with boundary in Riemannian manifolds obtained via a min-max construction, both in fixed and free boundary context. We start by considering a smooth, compact, oriented Riemmanian manifold ($\mathcal{M},\mathit{g}$) of dimension $n$ + 1 with a uniform convexity property (the principal curvatures of $\partial\mathcal{M}$ with respect to the inner normal have a uniform positive lower bound). In the spirit of an analogous approach presented in Colding-De Lellis (cf. [10]), we then construct generalized families of hypersurfaces in $\mathcal{M}$, with appropriate boundary conditions. To be more precise, in the fixed boundary setting this will basically mean that these hypersurfaces will have a fixed common boundary $\gamma$, which is an ($n$ - 1)-dimensional smooth, closed, oriented submanifold of $\partial\mathcal{M}$, and in the free boundary setting that their boundaries lie in $\partial\mathcal{M}$. Moreover, we will consider more general parameter spaces for these families. After constructing a suitable homotopy class of such families and assuming that it satisfies a certain gap, we can prove the existence of a nontrivial, embedded, minimal hypersurface (with a codimension 7 singular set) and corresponding boundary conditions. As a corollary, we consider a special case of two smooth, strictly stable surfaces bounding an open domain $A$ and meeting only in the common boundary $\gamma \subset \partial\mathcal{M}$, and show the existence of a homotopy class with the required energy gap. The main theorem then furnishes the existence of a third minimal surface.

Even triangulations ofn–dimensional pseudo-manifolds

This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n-manifolds, an interesting connection between very short hierarchies and even triangulations is observed.

Spinorial description of SU(3)- and G2-manifolds

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.

LIGHTLIKE HYPERSURFACES OF AN INDEFINITE KAEHLER MANIFOLD WITH A QUARTER-SYMMETRIC METRIC CONNECTION

Abstract. In this paper, we study lightlike hypersurfaces of an indefiniteKaehler manifold with a quarter-symmetric metric connection. We proveseveral classification theorems for such a lightlike hypersurface. 1. IntroductionA linear connection ∇¯ on a semi-Riemannian manifold (M,¯ ¯g) is said to bea quarter-symmetric connection if its torsion tensor T¯ satisfies(1.1) T¯(X,Y ) = π(Y )JX −π(X)JY,for any vector fields X and Y on M¯, where J is a (1,1)-type tensor field andπ is a 1-form associated with a non-vanishing smooth vector field ζ, which iscalled the torsion vector field of M¯, by π(X) = ¯g(X,ζ). Moreover, if ∇¯ satisfies∇¯¯g = 0, then it is called a quarter-symmetric metric connection.Quarter-symmetric metric connection was introduced by K. Yano and T.Imai [15], and then it have been studied by S. C. Rastogi [13, 14], D. Kamilyaand U. C. De [8], R. S. Mishra and S. N. Pandey [9], S. Golab [7] and others.On the other hand, N. Puˇsi´c [12], and J. Niki´c and Puˇsi´c [10] studied quarter-symmetric metric connections on Kaehler manifold.The theory of lightlike hypersurfaces is an important topic of research indifferential geometry due to its application in mathematical physics, especiallyin the general relativity. The study of such notion was initiated by Duggal andBejancu [3] and later studied by many authors (see recent results in two books[4, 6]). Although now we have lightlike version of a large variety of Riemann-ian submanifolds, the geometry of lightlike hypersurfaces of semi-Riemannianmanifolds with quarter-symmetric metric connections is hardly known.In this paper, we study lightlike hypersurfaces of an indefinite Kaehler man-ifold (M,¯ ¯g,J ) with a quarter-symmetric metric connection, in which the tensor

Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

We study lightlike hypersurfacesMof an indefinite generalized Sasakian space formM-(f1,f2,f3), with indefinite trans-Sasakian structure of type(α,β), subject to the condition that the structure vector field ofM-is tangent toM. First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type(α,β). Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

The first eigenvalue of Laplace-type elliptic operators induced by conjugate connections

According to Tashiro–Obata, on a Riemannian manifold (M, g) with its Ricci curvature bounded positively from below, the first eigenvalue of the Laplacian on functions satisfies a simple inequality in terms of the scalar curvature, and equality characterizes the Riemannian sphere. We discuss a similar inequality for a certain elliptic operator on a manifold with conjugate connections. As application we characterize hyperellipsoids in Blaschke’s unimodular-affine hypersurface theory.