The tangent-point energy for surfaces and its symmetric critical points |
Axel Wings |
Dienstag, 28. Januar 2025 14:15Uhr |
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We will prove the existence of several distinct surfaces of the same given genus that are critical points of the tangent-point energy. The first step of this proof is to pull the tangent-point energy into our comfort zone. The key idea of this step is to describe the surfaces by embeddings of a $2$D manifold $M$ into $\mathbb{R}^3.$ We will define the tangent-point energy on the set of $W^{s,q}$-embeddings, which is an open subset of the Banach space $W^{s,q}(M,\mathbb{R}^3).$ We will discuss this space and characterize the energy space in terms of this regularity. We will see that the tangent-point energy of each $W^{s,q}$-embedding is finite, and each surface with finite energy can be described by a $W^{s,q}$-embedding. Furthermore, we will show that the tangent-point energy is continuously Fréchet differentiable on this domain. Once we have reached this comfortable situation, we will study the energy landscape. By an application of Palais' principle of symmetric criticality and a symmetry argument, we will establish the claimed result. |