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Giuseppe Tinaglia

Publications and source records attributed to Giuseppe Tinaglia.

33 records · Page 2Linked to original sources
arXiv2015

One-sided curvature estimates for H-disks

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $\mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].

math.DG↗
arXiv2015

Limit lamination theorem for H-disks

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi in [8]. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in $\mathbb{R}^3$ with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi in [9] for minimal disks.

math.DG↗
arXiv2014

$C^{1,α}$-regularity for surfaces with $H$ in $L^p$

In this paper we prove several results on the geometry of surfaces immersed in $\mathbf R^3$ with small or bounded $L^2$ norm of $|A|$. For instance, we prove that if the $L^2$ norm of $|A|$ and the $L^p$ norm of $H$, $p>2$, are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded $L^2$ norm of $|A|$, not necessarily small, then such a disk is graphical away from its boundary, provided that the $L^p$ norm of $H$ is sufficiently small, $p>2$. These results are related to previous work of Schoen-Simon and Colding-Minicozzi.

math.DG↗
arXiv2012

Non-proper complete minimal surfaces embedded in H^2 x R

Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical translation or by a hyperbolic or parabolic screw motion. In particular, we construct a large family of non-proper complete minimal disks embedded in H^2 x R invariant by a vertical translation and a hyperbolic screw motion and whose importance is twofold. They have finite total curvature in the quotient of H^2 x R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R^3 satisfying the same properties. And they show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H^2 x R.

math.DG↗
arXiv2011

Curvature estimates for surfaces with bounded mean curvature

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.

math.DG↗
arXiv2010

Existence of regular neighborhoods for H-surfaces

In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in certain n-manifolds.

math.DG↗
arXiv2008

The number of constant mean curvature isometric immersions of a surface

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x: M \to R^3 of an oriented non-simply-connected surface with constant mean curvature H. We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of R^3, either finite or a circle. When it is a circle then, for the immersion x, every cycle in M has vanishing force and, when H is not 0, also vanishing torque. Our work generalizes a rigidity result for minimal surfaces to constant mean curvature surfaces. Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque.

math.DG↗
arXiv2008

The Dynamics Theorem for CMC surfaces in R^3

In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surfaces in minimal T-invariant sets of T(M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T-invariant set in T(M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.

math.DG↗
arXiv2008

The rigidity of embedded constant mean curvature surfaces

We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.

math.DG↗
arXiv2007

Curvature estimates for minimal surfaces with total boundary curvature less than 4π

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4π. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than 4πand which do not bound an orientable compact embedded minimal surface of genus greater than g, for any given g, is open in the C^{2,α} topology.

math.DG↗
arXiv2005

A generalization of Rado's Theorem for almost graphical boundaries

In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is "almost graphical" in some sense, that the surface must be graphical once we move sufficiently far from the boundary.

math.DG↗
arXiv2004

Multi-valued graphs in embedded constant mean curvature disks

In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of $|A|^2$, where $|A|^2$ is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for minimal surfaces.

math.DG↗