On spectral geometry of minimal parallel submanifolds

We consider the problem of characterizing some minimal submanifolds using the spectrum⁰Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCP ⁿ, 3≤n≤6, by their⁰Spec in the class of all compact totally real minimal submanifolds ofCP ⁿ. Moreover, we characterize the Clifford torus by its⁰Spec in the class of all compact minimal submanifolds of the Euclidean sphereS ⁿ⁺¹(1). Authors supported by funds of the University of Lecce and the M.U.R.S.T.     

n-dimensional totally real minimal submanifolds of CP n

Let CP ⁿ be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CP ⁿ . In this paper we prove the following results.

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Uniform convexity and smoothness,and their applications in Finsler geometry

We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.     

Isoperimetric comparison theorems for manifolds with density

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r ^α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.     

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H ¹ ₀(Γ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.     

Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs

We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs. G. Arena is supported by MIUR (Italy), by INDAM and by University of Trento. R. Serapioni is supported by MIUR (Italy), by GALA project of the Sixth Framework Programme of European Community and by University of Trento.     

Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II

In this article, the author proves that a simple closed polygon can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles at its N + 3 vertices, has to be smaller than 6π.      

Hilbert geometry of polytopes

It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension. Supported by the Schweizerischer Nationalfonds grants SNF PP002-114715/1 and 200020-121506/1.     

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ?M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ?M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations ${\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkh?user, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations of the fundamental group π _g,l of a punctured Riemann surface Σ _g,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on . We show that the involution is induced by a form-reversing involution β defined on the quasi-Hamiltonian space . The fact that has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser (A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. ). The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained. Supported by the Japanese Society for Promotion of Science (JSPS).     

Rational convexity of non-generic immersed Lagrangian submanifolds

We prove that an immersed Lagrangian submanifold in C ⁿ with quadratic self-tangencies is rationally convex. This generalizes former results for the embedded and the immersed transversal cases.     

Finsler metrics with positive constant flag curvature

We showed that any reversible Finsler metric with positive constant flag curvature must be Riemannian. Received: 18 August 2008     

Examples of area-minimizing surfaces in the sub-Riemannian Heisenberg group $${\mathbb{H}^1}$$ with low regularity

We give new examples of entire area-minimizing t-graphs in the sub-Riemannian Heisenberg group . They are locally Lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained. Research supported by MEC-Feder grant MTM2007-61919.     

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S ₀. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S ₀. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S ₀ attached to each other by narrow necks and winding a large number of times around before closing up on themselves; and are topologically equivalent to .     

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

We consider graphs with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate

Torsional rigidity of submanifolds with controlled geometry

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P ^m with controlled radial mean curvature in ambient Riemannian manifolds N ⁿ with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in, e.g., Markvorsen and Palmer (Proc Lond Math Soc 93:253--272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion.     

Triply periodic minimal surfaces bounded by vertical symmetry planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz–Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. S. Fujimori was partially supported by JSPS Grant-in-Aid for Young Scientists (Start-up) 19840035. M. Weber’s material is based upon work for the NSF under Award No. DMS-0139476.     

Bending the helicoid

We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C ¹-convergence being any properly embedded C ^1,1-curve. By Meeks’ C ^1,1-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination \({\mathcal{L}}\) is a locally finite collection \(S({\mathcal{L}})\) of C ^1,1-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle \({\mathbb{S}}^1(1)\) in the (x ₁, x ₂)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H _n of finite total curvature which contain the circle. The complete minimal surfaces H _n contain embedded compact minimal annuli \(\overline{H}_n\) in closed compact neighborhoods N _n of the circle that converge as \(n \to \infty\) to \(\mathbb {R}^3 - x_3\) -axis. In this case, we prove that the \(\overline{H}_n\) converge on compact sets to the foliation of \(\mathbb {R}^3 - x_3\) -axis by vertical half planes with boundary the x ₃-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C ¹-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.