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Zero mean curvature surfaces
Monday, 02 September 2024
Time Speaker Title Resources
10:00 to 11:30 John Loftin (Rutgers University, New Jersey, USA) Equivariant Minimal Surfaces in the Symmetric Spaces via Higgs Bundles (Lecture 1)

I will first consider the geometry of the complex hyperbolic plane and immersed surfaces therein, in particular the cases of Lagrangian and complex surfaces. The complex surfaces are all minimal, but there are many others as well. As it is a symmetric space, the more general case of harmonic maps from a Riemann surface into the complex hyperbolic plane naturally generates holomorphic data of a Higgs bundle. We impose a compactness condition to relate our study of minimal surfaces to Higgs bundles. Let S be a closed Riemann surface of genus at least 2. Consider then harmonic immersions of the universal cover of S into the complex hyperbolic plane which are equivariant with respect to some representation of the fundamental group of S into the group P U(2, 1) of holomorphic isometries of the complex hyperbolic plane. In this case, the nonlinear Hodge correspondence applies and thus there is a (poly-)stable Higgs bundle over S. In the standard case of the group GL(n, C), this consists of a holomorphic vector bundle E of rank n over S together with the Higgs field: a holomorphic section Φ of End(E) ⊗ K. The cases of other groups such as our group of isometries can typically be addressed as modifications of this case. The moduli theory of Higgs bundles over Riemann surfaces is in general rather complicated, as in particular it depends on an arbitrary vector bundle E. In important special cases we can determine the Higgs bundle in terms of simpler objects. (This approach goes back to Hitchin’s original work in parametrizing Teichmuller space with quadratic differentials via Higgs bundles.) The minimal surface condition (that the harmonic map is conformal) provides enough information to classify all such equivariant minimal surfaces into the complex hyperbolic plane, as well as into a few other cases of minimal surfaces into real hyperbolic spaces of degree 3 and 4. This is based on joint work from 2019 with Ian McIntosh, as well as work of Alessandrini-Li.
12:00 to 13:00 Franz E. Weber (University of Zurich (UZH), Zurich, Switzerland) 3D-printed bone substitutes: From pores to adaptive density minimal surface microarchitecture (Lecture1)

Introduction: In the last decades, advances in bone tissue engineering mainly based on osteoinduction and on stem cell research. Only recently, new efforts focused on the micro- and nanoarchitecture of bone substitutes to improve and accelerate bone regeneration. By the use of additive manufacturing, diverse microarchitectures were tested to identify the ideal pore size [1], the ideal filament distance and diameter [2], or light-weight microarchitecture [3], for osteoconduction to minimize the chance for the development of non-unions. Overall, the optimal microarchitecture doubled the efficiency of scaffold-based bone regeneration without the need for growth factors or cells. Another focus is on bone augmentation, a procedure mainly used in the dental field.

Methods: For the production of scaffolds, we applied the CeraFab 7500 from Lithoz, a lithographybased additive manufacturing machine. Hydroxyapatite-based and tri-calcium-phosphate-based scaffolds were produced with Lithoz TCP 300 or HA 400 slurries.

Results: The histomorphometric analysis revealed that bone ingrowth was significantly increased with pores between 0.7-1.2 mm in diameter. Best pore-size for bone augmentation was 1.7 mm in diameter. Therefore, pore-based microarchitectures for osteoconduction and bone augmentation are different. Moreover, microporosity appeared to be a strong driver of osteoconduction and influenced osteoclastic degradation for tri-calcium phosphate-based scaffolds. For hydroxyapatitebased scaffolds, however, microporosity appears to influence osteoconductivity to a lesser extent. Osteoclasts were able to degrade hydroxyapatite-based scaffolds irrespective of nanoarchitecture but tri-calcium phosphate-based scaffolds only at high and moderate levels of microporosity. The evaluation of the gene expression profiles at early bone healing leading to osteoconduction revealed that a reduction of the filament size from 1.25 mm to 0.5 mm yielded in differentiation of mesenchymal stem cells towards osteoclasts, enhanced angiogenesis and increased osteoconduction.

Discussion: Micro- and nanoarchitectures are key driving forces for osteoconduction and bone augmentation. A variety of microarchitectures can be realized by additive manufacturing. We identified optimized lattice, pore, filament and adjusted density minimal surface architectures for bony bridging and bone augmentation purposes. Based on these new results additive manufacturing appears as the tool of choice to produce personalized bone tissue engineering scaffolds to be used in cranio-maxillofacial surgery, dentistry, and orthopedics.

References:

  1. Ghayor et al Weber FE (2021) The optimal microarchitecture of 3D-printed β-TCP bone substitutes for vertical bone augmentation differs from that for osteoconduction. Materials and Design 204 (2021) 109650
  2. Guerrero J, et al Weber FE, (2023) Influence of Scaffold’s Microarchitecture on Angiogenesis and Regulation of Cell Differentiation During the Early Phase of Bone Healing: a Transcriptomics and Histological Analysis. Int J Bioprint, 9(1): 0093.
  3. Maevskaia et al Weber FE (2022) 3D-printed hydroxyapatite bone substitutes designed by a novel periodic minimal surface algorithm are highly osteoconductive.
14:30 to 16:00 Shoichi Fujimori (Hiroshima University, Higashihiroshima, Japan) Zero mean curvature surfaces in Lorentz Minkowski spaces (Lecture1)

First Talk (90 minutes) I would like to give a brief introduction of spacelike hypersurfaces and maximal hypersurfaces in Minkowski space. Then I would like to introduce maximal surfaces with singularities in Minkowsiki 3-space and give examples.
16:30 to 17:30 Randall Kamien (University of Pennsylvania (UPenn), Philadelphia, USA) Minimal Surfaces in Diblock Copolymers and geometric scaffolds in materials science (Lecture 1)- ONLINE

Outline of Topics:

  • Minimal Surfaces in Diblock Copolymers: the P,G, and D triply periodic minimal surfaces.
  • Smectic Liquid Crystals: the simplest crystals and they are built from surfaces.
  • Using minimal surfaces to knit: geometric scaffolds in materials science.
Tuesday, 03 September 2024
Time Speaker Title Resources
10:00 to 11:30 John Loftin (Rutgers University, New Jersey, USA) Equivariant Minimal Surfaces in the Symmetric Spaces via Higgs Bundles. (Lecture 2)

I will first consider the geometry of the complex hyperbolic plane and immersed surfaces therein, in particular the cases of Lagrangian and complex surfaces. The complex surfaces are all minimal, but there are many others as well. As it is a symmetric space, the more general case of harmonic maps from a Riemann surface into the complex hyperbolic plane naturally generates holomorphic data of a Higgs bundle. We impose a compactness condition to relate our study of minimal surfaces to Higgs bundles. Let S be a closed Riemann surface of genus at least 2. Consider then harmonic immersions of the universal cover of S into the complex hyperbolic plane which are equivariant with respect to some representation of the fundamental group of S into the group P U(2, 1) of holomorphic isometries of the complex hyperbolic plane. In this case, the nonlinear Hodge correspondence applies and thus there is a (poly-)stable Higgs bundle over S. In the standard case of the group GL(n, C), this consists of a holomorphic vector bundle E of rank n over S together with the Higgs field: a holomorphic section Φ of End(E) ⊗ K. The cases of other groups such as our group of isometries can typically be addressed as modifications of this case. The moduli theory of Higgs bundles over Riemann surfaces is in general rather complicated, as in particular it depends on an arbitrary vector bundle E. In important special cases we can determine the Higgs bundle in terms of simpler objects. (This approach goes back to Hitchin’s original work in parametrizing Teichmuller space with quadratic differentials via Higgs bundles.) The minimal surface condition (that the harmonic map is conformal) provides enough information to classify all such equivariant minimal surfaces into the complex hyperbolic plane, as well as into a few other cases of minimal surfaces into real hyperbolic spaces of degree 3 and 4. This is based on joint work from 2019 with Ian McIntosh, as well as work of Alessandrini-Li.
12:00 to 13:00 Franz E. Weber (University of Zurich (UZH), Zurich, Switzerland) 3D-printed bone substitutes with triply periodic minimal surface microarchitectures (Lecture 2)

Introduction: Additive manufacturing or 3D printing are key methodologies to produce libraries of bone substitutes to test them to identify highly osteoconductive microarchitectures for bone defects or bone augmentation. Bone is a lightweight, high strength structure and resembles in its trabecular microarchitecture a gothic style. TPMS architectures are also lightweight and high strength. Therefore, we produced triply periodic minimal surface (TPMS) lightweight-based scaffolds based on three different algorithms and tested them in a cranial defect and a bone augmentation model in rabbits. 8 Methodology For the production of scaffolds, we applied the CeraFab 7500 from Lithoz, a lithography-based additive manufacturing machine and studied tri-calcium phosphate- based and hydroxyapatite-based scaffolds. As in vivo test model, we used a calvarial defect and a bone augmentation model in rabbits. Histomorphometry revealed that all generatively produced structures were well osseointegrated into the surrounding bone and induced bone augmentation. The histomorphometric analysis, based solely on the middle section combined with microCT analysis showed that for triply periodic minimal surface lightweight microarchitectures, gyroid- and diamond- microarchitecture performs well in bone augmentation and cranial defect models.

Results: In essence, we have identified the optimal triply periodic lightweight microarchitecture for cranial defects (1) and for bone augmentation purposes needed for the placement of dental implants (2). We learned before that the optimal pore-based and filament-based microarchitecture for bone augmentations differ from the best for the treatment of defects. For TPMS-based microarchitectures, however, diamond and especially gyroid microarchitectures are optimal for bone augmentation and osteoconduction. Moreover, we saw that additive manufacturing appears as a promising tool for the production of personalized bone substitutes to be used in craniomaxillofacial surgery, dentistry, and orthopedics. The addition of exosomes to further enhance osteoconduction and bone augmentation did not yield better results (3).

Discussion: Micro- and nanoarchitectures are key driving forces for osteoconduction and bone augmentation. A variety of microarchitectures can be realized by additive manufacturing. We identified optimized TPMS microarchitectures for bony bridging and bone augmentation purposes. Based on these new results additive manufacturing appears as the tool of choice to produce personalized bone tissue engineering scaffolds to be used in cranio-maxillofacial surgery, dentistry, and orthopedics.

References:

  1. Maevskaia et al Weber FE (2023) TPMS-based scaffolds for bone tissue engineering: a mechanical, in vitro and in vivo study.Tissue Engineering Part A.
  2. Maevskaia E, et al Weber FE, TPMS Microarchitectures for Vertical Bone Augmentation and Osteoconduction: An In Vivo Study Materials 2024, 17, 2533.
  3. Maevskaia E, et al Weber FE, Functionalization of Ceramic Scaffolds with Exosomes from Bone Marrow Mesenchymal Stromal Cells for Bone Tissue Engineering. Int. J. Mol. Sci. 2024 25, 3826.
14:30 to 16:00 Shoichi Fujimori (Hiroshima University, Higashihiroshima, Japan) Zero mean curvature surfaces in Lorentz Minkowski spaces (Lecture 2)

Second Talk (90 minutes) I would like to focus a construction of nonorientable maximal surfaces. Then I would like to give some classification theorems.
16:30 to 17:30 Randall Kamien (University of Pennsylvania (UPenn), Philadelphia, USA) Minimal Surfaces in Diblock Copolymers and geometric scaffolds in materials science (Lecture 2)- ONLINE

Outline of Topics:

  • Minimal Surfaces in Diblock Copolymers: the P,G, and D triply periodic minimal surfaces.
  • Smectic Liquid Crystals: the simplest crystals and they are built from surfaces.
  • Using minimal surfaces to knit: geometric scaffolds in materials science.
Wednesday, 04 September 2024
Time Speaker Title Resources
09:30 to 11:00 John Loftin (Rutgers University, New Jersey, USA) Equivariant Minimal Surfaces in the Symmetric Spaces via Higgs Bundles. (Lecture 3)

I will first consider the geometry of the complex hyperbolic plane and immersed surfaces therein, in particular the cases of Lagrangian and complex surfaces. The complex surfaces are all minimal, but there are many others as well. As it is a symmetric space, the more general case of harmonic maps from a Riemann surface into the complex hyperbolic plane naturally generates holomorphic data of a Higgs bundle. We impose a compactness condition to relate our study of minimal surfaces to Higgs bundles. Let S be a closed Riemann surface of genus at least 2. Consider then harmonic immersions of the universal cover of S into the complex hyperbolic plane which are equivariant with respect to some representation of the fundamental group of S into the group P U(2, 1) of holomorphic isometries of the complex hyperbolic plane. In this case, the nonlinear Hodge correspondence applies and thus there is a (poly-)stable Higgs bundle over S. In the standard case of the group GL(n, C), this consists of a holomorphic vector bundle E of rank n over S together with the Higgs field: a holomorphic section Φ of End(E) ⊗ K. The cases of other groups such as our group of isometries can typically be addressed as modifications of this case. The moduli theory of Higgs bundles over Riemann surfaces is in general rather complicated, as in particular it depends on an arbitrary vector bundle E. In important special cases we can determine the Higgs bundle in terms of simpler objects. (This approach goes back to Hitchin’s original work in parametrizing Teichmuller space with quadratic differentials via Higgs bundles.) The minimal surface condition (that the harmonic map is conformal) provides enough information to classify all such equivariant minimal surfaces into the complex hyperbolic plane, as well as into a few other cases of minimal surfaces into real hyperbolic spaces of degree 3 and 4. This is based on joint work from 2019 with Ian McIntosh, as well as work of Alessandrini-Li.
11:30 to 13:00 Nathaniel Sagman (University of Luxembourg, Luxembourg, Germany) Minimal surfaces in symmetric spaces and Labourie’s Conjecture (Lecture 1)- ONLINE

I will discuss geometric and analytic aspects of the theory of equivariant harmonic maps and (branched) minimal immersions from surfaces into Riemannian symmetric spaces of non-compact type, such as SL(n,R)/SO(n,R). We’ll also highlight certain research directions and share open problems. First, I will provide an overview of the basic existence and uniqueness theorems on harmonic maps due to Donaldson and Corlette, and I will explain how harmonic maps to symmetric spaces give rise to holomorphic objects called Higgs bundles (and moreover play a prominent role in the non-abelian Hodge correspondence). Next, I will introduce Hitchin representations and the surrounding minimal surface existence theory, and Labourie’s conjecture about the uniqueness of equivariant minimal immersions for Hitchin representations. I will then detail the state of the art on high energy harmonic maps to symmetric spaces, and explain how to use this theory to construct unstable equivariant minimal surfaces for Hitchin representations, which lead to counterexamples to Labourie’s conjecture.

Reference:
Thursday, 05 September 2024
Time Speaker Title Resources
10:00 to 11:30 Shoichi Fujimori (Hiroshima University, Higashihiroshima, Japan) Zero mean curvature surfaces in Lorentz Minkowski spaces (Lecture 3)

Third Talk (90 minutes) I would like to give a brief introduction of zero mean curvature surfaces as the analytic extension of maximal surfaces to timelike minimal surfacces. Then I would like to give numerous examples of zero mean curvature surfaces and discuss some properties of them such as embeddedness and symmetries.
12:00 to 13:00 - Short Talks/ Student's Talks
14:30 to 16:00 Nathaniel Sagman (University of Luxembourg, Luxembourg, Germany) Minimal surfaces in symmetric spaces and Labourie’s Conjecture (Lecture 2)- ONLINE

I will discuss geometric and analytic aspects of the theory of equivariant harmonic maps and (branched) minimal immersions from surfaces into Riemannian symmetric spaces of non-compact type, such as SL(n,R)/SO(n,R). We’ll also highlight certain research directions and share open problems. First, I will provide an overview of the basic existence and uniqueness theorems on harmonic maps due to Donaldson and Corlette, and I will explain how harmonic maps to symmetric spaces give rise to holomorphic objects called Higgs bundles (and moreover play a prominent role in the non-abelian Hodge correspondence). Next, I will introduce Hitchin representations and the surrounding minimal surface existence theory, and Labourie’s conjecture about the uniqueness of equivariant minimal immersions for Hitchin representations. I will then detail the state of the art on high energy harmonic maps to symmetric spaces, and explain how to use this theory to construct unstable equivariant minimal surfaces for Hitchin representations, which lead to counterexamples to Labourie’s conjecture.

Reference:
16:30 to 17:30 L Mahadevan (Harvard University, Cambridge, USA) Soft surfaces bounded by soft filaments: from the Euler-Plateau problem to morphogenesis- ONLINE
Friday, 06 September 2024
Time Speaker Title Resources
10:00 to 11:30 Nathaniel Sagman (University of Luxembourg, Luxembourg, Germany) Minimal surfaces in symmetric spaces and Labourie’s Conjecture (Lecture 3)- ONLINE

I will discuss geometric and analytic aspects of the theory of equivariant harmonic maps and (branched) minimal immersions from surfaces into Riemannian symmetric spaces of non-compact type, such as SL(n,R)/SO(n,R). We’ll also highlight certain research directions and share open problems. First, I will provide an overview of the basic existence and uniqueness theorems on harmonic maps due to Donaldson and Corlette, and I will explain how harmonic maps to symmetric spaces give rise to holomorphic objects called Higgs bundles (and moreover play a prominent role in the non-abelian Hodge correspondence). Next, I will introduce Hitchin representations and the surrounding minimal surface existence theory, and Labourie’s conjecture about the uniqueness of equivariant minimal immersions for Hitchin representations. I will then detail the state of the art on high energy harmonic maps to symmetric spaces, and explain how to use this theory to construct unstable equivariant minimal surfaces for Hitchin representations, which lead to counterexamples to Labourie’s conjecture.

Reference:
12:00 to 13:00 - Short Talks/ Student's Talks
16:30 to 17:30 Rahul Kumar Singh (Indian Institute of Technology, Patna (IITP), India) CMC surfaces of revolution in E3 1 and Weierstass-℘ functions

It is a well-known fact that in the class of regular non-zero constant mean curvature (CMC) surfaces in the Euclidean space, spheres and the right circular cylinders are the only examples of CMC surfaces which are algebraic. In this talk, first we will show for every spacelike CMC surface of revolution (except spacelike cylinders and standard hyperboloids), which is either an unduloid or a nodoid, in the Lorentz-Minkowski space E 3 1 , there is an associated Weierstrass-℘ function. Next, using this association, we will show unduloid and nodoid cannot be algebraic and hence concluding only spacelike cylinders and standard hyperboloids are algebraic.