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William H Meeks - One of the best experts on this subject based on the ideXlab platform.
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local Removable Singularity theorems for minimal laminations
Journal of Differential Geometry, 2016Co-Authors: William H Meeks, Joaquin Perez, Antonio RosAbstract:In this paper we prove a local Removable Singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This Removable Singularity theorem is the key result used in our proof that a complete, embedded minimal surface in R 3 with quadratic decay of curvature has nite total curvature.
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The classification of CMC foliations of R3 and S3 with countably many singularities
2014Co-Authors: William H Meeks, Pérez Antonio RosAbstract:In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak H-laminations (with H ∈ R constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mean curvature from leaf to leaf) of a compact Riemannian three-manifoldN with boundary solely in terms of a bound of the absolute sectional curvature of N and of the distance to the boundary of N. We then apply these results to classify weak CMC foliations of R3 and S3 with a closed countable set of singularities
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a survey on classical minimal surface theory
2012Co-Authors: William H Meeks, Joaquin PerezAbstract:Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the focus of the account. The proof of the classification depends on the work of many currently active leading mathematicians, thus making contact with much of the most important results in the field. Through the telling of the story of the classification of minimal planar domains, the general mathematician may catch a glimpse of the intrinsic beauty of this theory and the authors' perspective of what is happening at this historical moment in a very classical subject. This book includes an updated tour through some of the recent advances in the theory, such as Colding-Minicozzi theory, minimal laminations, the ordering theorem for the space of ends, conformal structure of minimal surfaces, minimal annular ends with infinite total curvature, the embedded Calabi-Yau problem, local pictures on the scale of curvature and topology, the local Removable Singularity theorem, embedded minimal surfaces of finite genus, topological classification of minimal surfaces, uniqueness of Scherk singly periodic minimal surfaces, and outstanding problems and conjectures.
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Structure theorems for singular minimal laminations
2008Co-Authors: William H Meeks, Pérez Antonio RosAbstract:In this paper, we apply our local Removable Singularity theorem and local struc-ture theorems for embedded minimal surfaces and minimal laminations in R3 proven in [16, 15], to obtain global structure theorems for certain possibly singular minimal laminations of R3. We will use Theorems 1.3 and Theorem 1.6 below in [14] to prove that a complete, embedded minimal surface in R3 with finite genus and a countable number of ends is proper. Theorem 1.6 will also be applied in [13] to obtain bounds on the index and the topology of complete, embedded minimal surfaces of fixed genus and finite topology in R3
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The structure of stable minimal surfaces near a Singularity
2006Co-Authors: William H MeeksAbstract:estimates, minimal lamination, Removable Singularity. 1 Introduction. Recently, Meeks, Perez and Ros [5] proved the following remarkable local Removable sin-gularity result for a minimal lamination of a Riemannian three-manifold N: If S ⊂ N is a closed countable set and L is a minimal lamination of N − S which satisfies in a punctured neighborhood W of each isolated point p of S a curvature estimate of the for
Guofang Wang - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Dirac equations on Riemann surfaces
2016Co-Authors: Qun Chen, Guofang Wang, J. JostAbstract:Abstract. We develop analytical methods for nonlinear Dirac equations. Ex-amples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy reg-ularity and Removable Singularity theorems and energy identities for solutions
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Dirac-harmonic maps
2015Co-Authors: Qun Chen, Guofang WangAbstract:Abstract. We introduce a functional that couples the nonlinear sigma model with a spinor field: L = M [|dφ|2 + (ψ,D/ψ)]. In two dimensions, it is confor-mally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a Removable Singularity theorem. 1
Oliveira G - One of the best experts on this subject based on the ideXlab platform.
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SU(2)2-invariant G2-instantons
'Springer Science and Business Media LLC', 2018Co-Authors: Jd Lotay, Oliveira GAbstract:We initiate the systematic study of G2-instantons with SU(2)2-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R4×S3 with its two explicitly known distinct holonomy G2 metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of G2-instantons where “bubbling” and “Removable Singularity” phenomena occur in the limit
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SU(2)² -invariant G₂ -instantons
2018Co-Authors: Jd Lotay, Oliveira GAbstract:We initiate the systematic study of G₂-instantons with SU(2)² -symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on R⁴ x S³ with its two explicitly known distinct holonomy G₂ metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We alsogive an explicit example of sequences of G₂-instantons where “bubbling” and “Removable Singularity” phenomena occur in the limit
Qun Chen - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Dirac equations on Riemann surfaces
2016Co-Authors: Qun Chen, Guofang Wang, J. JostAbstract:Abstract. We develop analytical methods for nonlinear Dirac equations. Ex-amples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy reg-ularity and Removable Singularity theorems and energy identities for solutions
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Dirac-harmonic maps
2015Co-Authors: Qun Chen, Guofang WangAbstract:Abstract. We introduce a functional that couples the nonlinear sigma model with a spinor field: L = M [|dφ|2 + (ψ,D/ψ)]. In two dimensions, it is confor-mally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a Removable Singularity theorem. 1
Ros Antonio - One of the best experts on this subject based on the ideXlab platform.
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Structure theorems for singular minimal laminations
2016Co-Authors: Meeks Iii, William H., Perez Joaquin, Ros AntonioAbstract:We apply the local Removable Singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two global structure theorems will be applied in forthcoming papers to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in $\mathbb{R}^3$, and to prove that a complete, embedded minimal surface in $\mathbb{R}^3$ with finite genus and a countable number of ends is proper.Comment: 43 pages, 7 figures. Bibliography updated and reorganized the pape
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The classification of CMC foliations of $\mathbb{R}^3$ and $\mathbb{S}^3$ with countably many singularities
2014Co-Authors: Meeks Iii, William H., Perez Joaquin, Ros AntonioAbstract:In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak $H$-laminations (with $H\in \mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mean curvature from leaf to leaf) of a compact Riemannian three-manifold $N$ with boundary solely in terms of a bound of the absolute sectional curvature of $N$ and of the distance to the boundary of $N$. We then apply these results to classify weak CMC foliations of $\mathbb{R}^3$ and $\mathbb{S}^3$ with a closed countable set of singularities.Comment: 38 pages, 5 figure
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Local Removable Singularity theorems for minimal laminations
2013Co-Authors: Meeks Iii, William H., Perez Joaquin, Ros AntonioAbstract:In this paper we prove a local Removable Singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This Removable Singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature has finite total curvature.Comment: 41 pages, 8 figure