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1. Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

   https://doi.org/10.1007/s00220-021-04303-8  

   Huang, Jiaxi ; Tataru, Daniel ( January 2022 , Communications in Mathematical Physics)

   The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$$R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$$d\ge 4$.

    

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2. Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

   https://doi.org/10.1007/s12220-024-01652-3  

   Bauer, Martin ; Heslin, Patrick ; Maor, Cy ( May 2024 , The Journal of Geometric Analysis)

   We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order$$q\in [0,\infty )$$$q\in [0,\infty )$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if$$q>1/2$$$q>1/2$. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if$$q>3/2$$$q>3/2$, whereas if$$q<3/2$$$q<3/2$then finite-time blowup may occur. The geodesic completeness for$$q>3/2$$$q>3/2$is obtained by proving metric completeness of the space of$$H^q$$$H^q$-immersed curves with the distance induced by the Riemannian metric.

    

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3. Local Well-Posedness of the Skew Mean Curvature Flow for Small Data in $$d\geqq 2$$ Dimensions

   https://doi.org/10.1007/s00205-023-01952-y  

   Huang, Jiaxi ; Tataru, Daniel ( January 2024 , Archive for Rational Mechanics and Analysis)

   The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$$R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$$d\geqq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$$d\geqq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.

    

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4. Size dependence- and induced transformations- of fractional quantum Hall effects under tilted magnetic fields

   https://doi.org/10.1038/s41598-022-22812-x  

   Wijewardena, U. Kushan ; Nanayakkara, Tharanga R. ; Kriisa, Annika ; Reichl, Christian ; Wegscheider, Werner ; Mani, Ramesh G. ( November 2022 , Scientific Reports)

   Two-dimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electron-spin, is removed by a small Zeeman spin splitting,$$g \mu _B B$$$g{\mu }_{B}B$, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a re-ordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$\nu$$$\nu$for the$$R_{xx}$$$R_{\mathrm{xx}}$minimum, e.g., from$$\nu = 11/7$$$\nu =11/7$to$$\nu = 8/5$$$\nu =8/5$, and a corresponding change in the$$R_{xy}$$$R_{\mathrm{xy}}$, e.g., from$$R_{xy}/R_{K} = (11/7)^{-1}$$$R_{\mathrm{xy}}/R_K={(11/7)}^{-1}$to$$R_{xy}/R_{K} = (8/5)^{-1}$$$R_{\mathrm{xy}}/R_K={(8/5)}^{-1}$, with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$\nu = 4/3$$$\nu =4/3$and$$\nu = 7/5$$$\nu =7/5$resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$R_{xy}$$$R_{\mathrm{xy}}$at the$$R_{xx}$$$R_{\mathrm{xx}}$minima- the latter occurring for$$\nu = 4/3, 7/5$$$\nu =4/3,7/5$and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu$$$\nu$and$$R_{xy}$$$R_{\mathrm{xy}}$, but also the tilt- or Zeeman-energy-dependent- crossover between distinct FQHE associated with different Hall resistances.

    

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5. On Some Non-Rigid Unit Distance Patterns

   https://doi.org/10.1007/s00454-023-00503-2  

   Frankl, Nóra ; Woodruff, Dora ( July 2023 , Discrete & Computational Geometry)

   A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius$$1/{\sqrt{2}}$$$1/\sqrt{2}$. We also consider a similar problem about 3-regular unit distance graphs in $$\mathbb {R}^3$$$R^3$.

    

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