Steffen Froehlich
We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension
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We establish area bounds for two-dimensional immersions in R^3 and R^n. Namely, for μ-stable immersions in R^3 (R^n), for graphs in $\mathbb R^3$ which solve quasilinear equations in divergence form, and for graphs which are critical for Fermat-type variational problems in R^n.
We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.
In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimensions greater than two. As an application we include an a priori estimate for the corresponding torsion coefficients in arbitrary codimensions.
Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates
本文通过定量估计方法，证明了二维等距浸入的渐近刚性，采用欧几里得方法获得简单证明，结果具有独立价值。
Rigidity of codimension-1 isometric immersions in complete manifolds
研究了完备流形中余维数为1的等距浸入的渐进行为，通过弹性能量趋于零的浸入序列，证明其子序列收敛于等距浸入，扩展了Alpern等人结果，避免了Young测度的使用。
A Nash-Kuiper theorem for isometric immersions in a high codimension
本文研究高余维等距浸入问题，在局部情形下将短浸入逼近的Hölder正则性从θ∈(0,1/(n+2))改进到奇数维的θ∈(0,1/n)和偶数维的θ∈(0,1/(n+1))，并建立了显式C^{1}估计。
Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions
本文改进了 $C^{1,θ}$ 等距浸入的最优指数，通过凸积分方案和频率尺度分析，实现了对短浸入的均匀近似。
We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showing that extrinsic curvature, defined by a suitable pull-back of the volume form on the $n$-sphere via the Gauss map, is identical to intrinsic curvature, defined by the Pfaffian form. This latter fact is stated in form of an integral identity for the Brouwer degree of the Gauss m…
We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min\{2k, n\}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Ampère equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1)\times (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min\{2k, n\}$.
We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function, and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.
A geometric rigidity estimate for codimension-1 immersions into spheres
本文针对球面中余维数一的浸入，建立了由拉伸加弯曲能量控制的定量刚性估计，无需先验基本形式界，并通过法向延伸与局部化拼接证明。
We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling scenarii. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12π$, only three of them may occur as conformal images of bubbles of Willmore immersions.
Existence of disks with prescribed mean curvature and contact angle in arbitrary codimension
本文在任意余维数下建立极小极大理论，证明在适当假设下存在具有给定平均曲率和接触角的非恒定分支浸入圆盘或球面，并在欧氏空间中给出自由边界圆盘及维数相关的莫尔斯指标界。
We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12π$, only three of them may occur as conformal images of bubbles in Willmore min-max problems.
We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(σ, tα)$, where $σ$ is a conformal structure on a topological surface $S$, and $αdz^2$ is a holomorphic quadratic differential on the surface $(S,σ)$. We show that, for each $t \in (0,τ_0)$ for some $τ_0 > 0$, depending only on $(σ, α)$, there are at least two minimal immersions of closed surface of prescribed second fundamental form $Re(tα)$ in the conformal structure $σ$. Moreover, for $t$ sufficiently large, there exists no such minimal immersion. Asymptotically, as $t \to 0$…
Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.
Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films
研究任意维度和余维下等距浸入系统的灵活性，通过证明可由特定正则性的精确等距浸入逼近，得到阈值灵活性指数\(\alpha_0\)，并应用于预应变薄膜能量缩放估计，确定其缩放指数。
Min-max theory and existence of H-spheres with arbitrary codimensions
本文通过极小极大理论，在闭黎曼流形中证明了具有指定平均曲率、受控莫尔斯指标和任意余维数的分支浸入2-球面的存在性，并在特定曲率条件下推广至所有平均曲率选择，进而构造平行平均曲率球面并部分解决同伦问题。
Another look at a notion of fractional mass in codimension two
研究闭黎曼流形上余维二电流的分数s-质量，通过能量最小化与指定雅可比约束定义，证明平坦拓扑下s-质量的等紧性和Γ-收敛，并建立s-调和映射的正则性与奇异集维数界。
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean cu…
In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescribed by a given $C^{1, α}$ ambient function $g$ and the ambient space is a general $(n+1)$-dimensional Riemannian manifold; we give general conditions that imply that the support of the varifold, away from a possible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$…
Stability of isometric immersions of hypersurfaces
本文研究了黎曼流形超曲面等距浸入在基本形式 $L^p$-扰动下的稳定性，利用 Young 测度方法将先前结果推广到一般目标流形，去除了常曲率假设，并在欧氏目标情形下给出了无先验界的定量稳定性结果。
Full flexibility of the Monge-Ampère system in codimension $d_*-d+1$
研究Monge-Ampère系统在余维数$d_*-d+1$中$\mathcal{C}^{1,α}$解的稠密性，扩展并推广了Lewicka(2022)和Inauen-Lewicka(2025)的结果，同时推广到更一般的余维数情况。
Euler Immersion
研究弹性体浸入不可压缩无粘欧拉流体的流体-结构相互作用问题，证明在速度匹配边界条件下解析空间的局部适定性，并给出速度与应力匹配边界条件同时施加时解不存在的反例。
Regularity for the prescribed vertical mean curvature system in the plane
本文证明了平面中指定垂直平均曲率系统的Lipschitz解在全域正则，余维数任意且平均曲率在$L^r(r>2)$中，加强并推广了先前关于Lawson-Osserman猜想的结果。
Calibration energy and mean curvature flow
本文引入校准能量量化定向浸入与校准几何的偏差，证明其在平均曲率流下的精确耗散恒等式，并应用于孤子刚性和二维永恒解的收敛性。
We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$.…
We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening …
Smoothability of $L^p$-connections on bundles and isometric immersions with $W^{2,p}$-regularity
该研究解决了低正则性联络的光滑逼近问题，并将其应用于刻画容许W^{2,p}等距浸入但不容许光滑等距浸入的黎曼流形度量，推进了等距浸入的正则性理论。
Initiated by the work of Uhlenbeck in late 1970s, we study questions about the existence, multiplicity and asymptotic behavior for minimal immersions of closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions. In particular, we determine when exactly the solution is unique and when multiple solutions appear. Moreover, we analyze in detail the asymptotic behavior of the solutions when (and how) blowing up might occur. Interestingly the blow-up analysis exhibit …
Existence and non-existence of minimal graphs
本文通过平均曲率流研究任意维数余维数下极小曲面系统的Dirichlet问题，证明了平均凸有界域上极小图的存在性，推广了Jenkins-Serrin判据，并构造了余维数2不可解的反例，还探讨了唯一性。
Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting
本文建立余维数为一时环面值映射的 $W^{1,1}$ 能量、BV 提升能量与 normal/integral 流 Plateau 最小质量之间的等价关系，并给出积分 Plateau 问题解的界。
In this overview report we generalize Erhard Heinz' curvature estimate for minimal graphs in R^3 to graphs in R^n of prescribed mean curvature. Secondly, we analyse these problems in the frame of the outer differential geometry which leads us to the notions of normal torsion and normal curvature for immersions in R^4.
We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is $g ν$, for a given continuous positive ambient function $g$, and where $ν$ denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearence of "hidden boundaries", that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that $g$ does not prescribe the mean curvature in the limit. As a special instance of a…
We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Ampère equation $\det D^2 u(z)=(1/ψ(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=φ$, for $ψ$ a smooth positive function and $φ$ a bounded lower semicontinuous function. We then prove…
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