made more stuff transparent

Author: Anthony David Gruber

_pages/about.md

@@ -142,7 +142,7 @@ Rigidity Results for Immersed Submanifolds
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 ### Parallel Codazzi Tensors with Submanifold Applications  [Preprint](https://arxiv.org/abs/2004.03103#){: .btn .btn--info .btn--small}{: .align-right}
-<img src="/images/equations.png" style="max-height: 225px; max-width: 250px; margin-right: 16px" align=left>  **Abstract:** A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of constant sectional curvature. In particular, it is shown that if $$M$$ has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then $$M$$ is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when $$M$$ is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.
+<img src="/images/equations.pdf" style="max-height: 225px; max-width: 250px; margin-right: 16px" align=left>  **Abstract:** A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of constant sectional curvature. In particular, it is shown that if $$M$$ has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then $$M$$ is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when $$M$$ is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.
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@@ -178,19 +178,19 @@ Curvature Functionals and p-Willmore Energy
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 ### Instability of Closed $$p$$-Elastic Curves in $$\mathcal{S}^2$$  [Preprint](https://arxiv.org/abs/2110.14778#){: .btn .btn--info .btn--small}{: .align-right}
-<img src="/images/spherical_curve_3-5.jpg" style="max-height: 180px; max-width: 180px; margin-right: 16px" align=left> **Abstract:** For $$p \in \mathbb{R}$$, we show that non-circular closed p-elastic curves in $$\mathbb{S}^2$$ exist only when $$p=2$$, in which case they are classical elastic curves, or when $$p\in(0,1)$$. In the latter case, we prove that for every pair of relatively prime natural numbers $$n$$ and $$m$$ satisfying $$m<2n<2\sqrt{m}$$, there exists a closed spherical $$p$$-elastic curve with non-constant curvature which winds around a pole $$n$$ times and closes up in m periods of its curvature. Further, we show that all closed spherical $$p$$-elastic curves for $$p\in(0,1)$$ are unstable as critical points of the p-elastic energy.
+<img src="/images/spherical_curve_3-5.pdf" style="max-height: 180px; max-width: 180px; margin-right: 16px" align=left> **Abstract:** For $$p \in \mathbb{R}$$, we show that non-circular closed p-elastic curves in $$\mathbb{S}^2$$ exist only when $$p=2$$, in which case they are classical elastic curves, or when $$p\in(0,1)$$. In the latter case, we prove that for every pair of relatively prime natural numbers $$n$$ and $$m$$ satisfying $$m<2n<2\sqrt{m}$$, there exists a closed spherical $$p$$-elastic curve with non-constant curvature which winds around a pole $$n$$ times and closes up in m periods of its curvature. Further, we show that all closed spherical $$p$$-elastic curves for $$p\in(0,1)$$ are unstable as critical points of the p-elastic energy.
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 (Joint with [Magdalena Toda](http://www.math.ttu.edu/~mtoda/) and [Álvaro Pámpano](https://www.math.ttu.edu/~apampano/index.html).)
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 ### On p-Willmore Disks with Boundary Energies  [Preprint](https://arxiv.org/abs/2110.14778#){: .btn .btn--info .btn--small}{: .align-right}
-<img src="/images/bubble.jpg" style="max-height: 180px; max-width: 180px; margin-right: 16px" align=left> **Abstract:** We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.
+<img src="/images/bubble.pdf" style="max-height: 180px; max-width: 180px; margin-right: 16px" align=left> **Abstract:** We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.
 <br><br>
 (Joint with [Magdalena Toda](http://www.math.ttu.edu/~mtoda/) and [Álvaro Pámpano](https://www.math.ttu.edu/~apampano/index.html).)
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 ### Regarding the Euler-Plateau Problem with Elastic Modulus  [Preprint](https://arxiv.org/abs/2010.00149#){: .btn .btn--info .btn--small}{: .align-right}
-<img src="/images/elastic_modulus.png" style="max-height: 180px; max-width: 180px; margin-right: 16px; margin-bottom: 10px" align=left>  **Abstract:** We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily planar domains bounded by area-constrained elasticae. The cases of topological genus zero with multiple boundary components and unrestricted genus with control on the geodesic torsion are also discussed, and sufficient conditions are given which establish the same conclusion in these cases.
+<img src="/images/elastic_modulus.pdf" style="max-height: 180px; max-width: 180px; margin-right: 16px; margin-bottom: 10px" align=left>  **Abstract:** We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily planar domains bounded by area-constrained elasticae. The cases of topological genus zero with multiple boundary components and unrestricted genus with control on the geodesic torsion are also discussed, and sufficient conditions are given which establish the same conclusion in these cases.
 <br><br>
 (Joint with [Magdalena Toda](http://www.math.ttu.edu/~mtoda/) and [Álvaro Pámpano](https://www.math.ttu.edu/~apampano/index.html).)
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