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1. An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

   https://doi.org/10.1093/imrn/rnae154  

   Duncan, Jonah_A J; Sire, Yannick; Spruck, Joel (August 2024, International Mathematics Research Notices)

   Abstract Suppose that $$\Sigma ^{n}\subset \mathbb{S}^{n+1}$$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $$\lambda _{1}$$ of the induced Laplace–Beltrami operator on $$\Sigma $$ satisfies $$\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$$, where $$a_{n}$$ and $$b_{n}$$ are explicit dimensional constants and $$\Lambda $$ is an upper bound for the length of the second fundamental form of $$\Sigma $$. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $$\lambda _{1} \geq \frac{n}{2}$$ without any further assumptions on $$\Sigma $$. 

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2. A CR Embedding Problem for an Algebraic Levi Non-degenerate Hypersurface into a Hyperquadric

   Huang, Xiaojun; Xiao, Ming (September 2018, Geometric Complex Analysis)

   We give a survey on the current developments of the embeddability problem of a Levi non-degenerate hypersurface into its model, i.e., hyerquadrics.We also formulate and study a local sums-of-squares problem, and make connections with the embeddability problem. 

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3. Skew and sphere fibrations

   https://doi.org/10.1090/tran/8953  

   Harrison, Michael (July 2023, Transactions of the American Mathematical Society)

   A great sphere fibration is a sphere bundle with total space S n S^n and fibers which are great k k -spheres. Given a smooth great sphere fibration, the central projection to any tangent hyperplane yields a nondegenerate fibration of R n \mathbb {R}^n by pairwise skew, affine copies of R k \mathbb {R}^k (though not all nondegenerate fibrations can arise in this way). Here we study the topology and geometry of nondegenerate fibrations, we show that every nondegenerate fibration satisfies a notion of Continuity at Infinity, and we prove several classification results. These results allow us to determine, in certain dimensions, precisely which nondegenerate fibrations correspond to great sphere fibrations via the central projection. We use this correspondence to reprove a number of recent results about sphere fibrations in the simpler, more explicit setting of nondegenerate fibrations. For example, we show that every germ of a nondegenerate fibration extends to a global fibration, and we study the relationship between nondegenerate line fibrations and contact structures in odd-dimensional Euclidean space. We conclude with a number of partial results, in hopes that the continued study of nondegenerate fibrations, together with their correspondence with sphere fibrations, will yield new insights towards the unsolved classification problems for sphere fibrations. 

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4. Influence of Size on the Fractal Dimension of Dislocation Microstructure

   https://doi.org/10.3390/met9040478  

   Cui, Yinan; Ghoniem, Nasr (April 2019, Metals)

   Three-dimensional (3D) discrete dislocation dynamics simulations are used to analyze the size effect on the fractal dimension of two-dimensional (2D) and 3D dislocation microstructure. 2D dislocation structures are analyzed first, and the calculated fractal dimension ( n 2 ) is found to be consistent with experimental results gleaned from transmission electron microscopy images. The value of n 2 is found to be close to unity for sizes smaller than 300 nm, and increases to a saturation value of ≈1.8 for sizes above approximately 10 microns. It is discovered that reducing the sample size leads to a decrease in the fractal dimension because of the decrease in the likelihood of forming strong tangles at small scales. Dislocation ensembles are found to exist in a more isolated way at the nano- and micro-scales. Fractal analysis is carried out on 3D dislocation structures and the 3D fractal dimension ( n 3 ) is determined. The analysis here shows that ( n 3 ) is significantly smaller than ( n 2 + 1 ) of 2D projected dislocations in all considered sizes. 

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5. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

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