论文标题
概括线性化的加倍方法，I：一般理论和新的最小表面和自我缩减器
Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers
论文作者
论文摘要
在本文的第一部分中，我们将NK早期工作中介绍的线性化加倍（LD）方法推广，证明了一般定理表明，如果$σ$是嵌入了Riemannian的三个manifold $（n，g）$的封闭式最小的表面，则在其jacobi utsisupt utitive utitive floce niim fornes in prom forefential utige niim fornes in Imim niim nim nim nim nim in $ ld。 $ \ breve {m} $类似于由许多小伴奏桥连接的两份$σ$，可以通过PDE胶合方法构建。 （$σ$上的ld解决方案$φ$是雅各比方程的单一解决方案，具有对数奇点，在构造中被链球菌桥取代。）我们还确定了该面积$ | \ breve {m} | of $ \ breve and Is catter Its and senter的第一个非繁琐术语。 消极的; $ | \ breve {m} | <2 | σ| $跟随。 我们通过首先构建Clifford Torus的新双打来证明定理的适用性。然后，我们在一般$（O（2）\ Times \ Mathbb {Z} _2）$的LD解决方案的第二部分族中构建。-对称背景$（σ，N，G）$。在第一部分中与定理相结合，这意味着为这种背景构建了新的最小双打。 （一般背景的构造保持打开状态。）这将我们的早期工作概括为$σ= \ mathbb {s}^2 \ subset n = \ mathbb {s}^3 $即使在这种情况下也提供新的构造。 在第三部分中，应用第一部分和第二部分的结果 - 适当地针对类cat骨和关键的catenoid进行了适当修改 - 我们通过使球形自我缩短器或愤怒的圆环增加一倍的平均曲率流动的新自我缩短器，新的完整嵌入式嵌入式最小弯曲的最小弯曲度在Euclidean the Space中的最小弯曲范围越来越多，并通过增长了较小的Euclidean unimil，并增加单位球通过将关键伴侣加倍加倍。
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Σ$, a minimal surface $\breve{M}$ resembling two copies of $Σ$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $φ$ on $Σ$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|\breve{M}|$ of $\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|\breve{M}| < 2 | Σ|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)\times \mathbb{Z}_2)$-symmetric backgrounds $(Σ, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Σ=\mathbb{S}^2 \subset N=\mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.
