The Total Gauss Curvature of a Three-Manifold Immersed in r 4

Half-space theorems for properly immersed surfaces in $\mathbb{R}^3$ with prescribed mean curvature

Motivated by the large ammount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as a prescribed function of its Gauss map. In order to achieve this purpose, we will study the behavior at infinity of a 1-parameter family of properly embedded annuli that are analogous to the usual minimal catenoids.

Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas.

On curvature of surfaces immersed in normed spaces

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand-Diguet-Puiseux are derived. A generalization of Weyl's formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtaine…

Type-II singularities of two-convex immersed mean curvature flow

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension $n\geq 3$ is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfa…

Total squared mean curvature of immersed submanifolds in a negatively curved space

Let $n\ge 2$ and $k\ge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $δ<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $λ_1(M)$ of $M$ satisfies $$λ_1(M)\le n\left(δ+\frac{1}{\operatorname{Vol} M}\int_M |H|^2 \operatorname{dvol}\right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an ope…

The Willmore Conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.

Degree Theory of Immersed Hypersurfaces

We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature; extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-χ(M)$, where $χ(M)$ is the Euler characteristic of $M$.

Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]).

Harmonic Mean Curvature Lines on Surfaces Immersed in R3

Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the product of the principal curvatures $k_1, k_2$ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by ${\mathcal K}/{\mathcal H}$, where $ {\mathcal H}=({k_1}+k_2)/2$ is the arithmetic mean curvature. That is, ${\mathcal K}/{\mathca…

A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.

Anisotropic curvature flow of immersed curves

We prove short-time existence of ϕ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.

Differential geometry of immersed surfaces in three-dimensional normed spaces

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the rel…

Properly immersed surfaces in hyperbolic 3-manifolds

We study complete finite topology immersed surfaces $Σ$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $Σ$ is bounded from above by $a$ and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $Σ$ must be proper in $N$ and its total curvature must be equal to $2πχ(Σ)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $Σ$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less …

维数不超过18的浸入环面的法曲率

Normal curvature of immersed tori of dimension at most $18$

该研究证明维数不超过18的浸入n维环面在单位球中存在点满足法曲率平方的球面积分下界，回答了Petrunin的问题，采用标量曲率障碍与共形拉普拉斯论证完成证明。

Helicoidal surfaces rotating/translating under the mean curvature flow

We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to 0 and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfa…

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.

Clifford结构、双勒让德曲面与外曲率

Clifford structures, bilegendrian surfaces, and extrinsic curvature

本文用Clifford代数建立统一框架，将黎曼与半黎曼3-流形中的常外曲率浸入曲面对应于酉丛中的双勒让德曲面，并完整分类了3-球面单位切丛中的完备与紧致浸入双勒让德曲面。

Complete negatively curved immersed ends in $\Bbb R^3$

This paper extends, in a sharp way, the famous Efimov's Theorem to immersed ends in $\real^3$. More precisely, let $M$ be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of $M$ into $\Bbb R^3$ satisfying that $\int_M |K|=+\infty$ and $K\le-κ<0$, where $κ$ is a positive constant and $K$ is the Gaussian curvature of $M$. In particular Efimov's Theorem holds for complete Hadamard immersed surfaces, whose Gaussian curvature $K$ is bounded away from zero outside a compact set.

On the Index of Constant Mean Curvature Hypersurfaces

In 1968, Simons introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S^{n+1}. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. Many such results established lower bounds for the index. More recently, however, mathematicians have generalized these results to hypersurfaces with constant mean curvature. In this paper, we consider hypersurfaces of constant mean curvature immersed into the sphere and give low…

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordström manifolds.

Mean curvature versus diameter and energy quantization

We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed curves, of all genus, in a compact locally conformally symplectic manifold.

有限指数常均曲率超曲面在低维空间中的研究

Finite index constant mean curvature hypersurfaces in low dimensions

本文证明在六维流形中，完备有限指数浸入的常均曲率超曲面要么是极小的要么是紧致的，扩展了低维结果，并完成六维正曲率空间中双侧完备弱稳定常均曲率超曲面的分类。

Co-dimension one area-minimizing currents with $C^{1,α}$ tangentially immersed boundary having Lipschitz co-oriented mean curvature

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has mean curvature $=h ν_{T}$ where $h$ is a Lipschitz scalar-valued function and $ν_{T}$ is the generalized outward pointing normal of $\partial T$ with respect to $T.$ We give a partial boundary regularity result for such currents $T.$ We show that near any point $x$ in the suppor…

Capillary surfaces: stability, index and curvature estimates

In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of $\mathbb{R}^3$ which is minimal or has a contact angle less than o…

Geometric Mean Curvature Lines on Surfaces Immersed in R3

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive) of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by $\sqrt {\mathcal K}$, which is the geometric mean curvature of the principal curvatures $ k_1, k_2$ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves}, where $ k_1 = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined …

On the umbilic set of immersed surfaces in three-dimensional space forms

We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.

Volume properties and rigidity on self-expanders of mean curvature flow

In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the $L$-stability operator and discuss the $L$-stability of some special self-expanders. Besides, we prove that the surfaces $Γ\times\…

Anisotropic curvature flow of immersed networks

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.

The Gauss map and second fundamental form of surfaces in a Lie group

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $\mathbb{S}^3$ and establish a correspondence between simply connected surfaces having extrinsic curvature $K$, $K$ different from 0 and -1, immersed in $\mathbb{S}^3$ with simply connected surfaces having non-vanishing extrinsic curvature immersed in the euclidean space $\mathbb{…

On the characterization of minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$ and $\widetilde{\rm PSL}_2(\mathbb{R},τ)$

It is known that a complete immersed minimal surface with finite total curvature in $\mathbb H^2\times\mathbb R$ is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg, 2006; Hauswirth, Nelli, Sa Earp and Toubiana, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in $\mathbb H^2\times\mathbb …

Parallel Codazzi tensors with submanifold applications

A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of constant sectional curvature. In particular, it is shown that if $M$ has nonnegative sectional curvature and admits a Codazzi tensor with "parallel mean curvature", then $M$ is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when $M$ is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

具有非负曲率流形中的子流形直径估计

Diameter estimates for submanifolds in manifolds with nonnegative curvature

研究了闭合连通流形在非负截面曲率流形中的内在直径估计，基于均曲率场积分，并扩展到带边的紧凸表面。

Lines of Mean Curvature on Surfaces Immersed in R3

Associated to oriented surfaces immersed in R^3 here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k_1, k_2 of the immersion, is positive. The leaves of the foliations are the lines of M- mean curvature, along which the normal curvature of the immersion is given by a function M = M(k_1,k_2) in [k_1, k_2], called a M- mean curvature, whose properties extend and unify those of the arithmetic H=(k_1+k_2)/2, the geometric K^{1/2} and harmonic K/H=((1/{k_1} + 1/{k_2})/2…

论三维空间形式中的拟完备 k-曲面

On quasicomplete $k$-surfaces in $3$-dimensional space-forms

本文引入拟完备性概念，证明在三维空间形式中常正外在曲率且满足 k>Max(0,-c) 的拟完备浸入曲面必为测地球面，从而完成该类曲面的完整分类。

Isometric deformations of minimal surfaces in $S^{4}$

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that for any compact immersed minimal surface in $S^{4}$ with nontrivial normal bundle there are at most finitely many noncongruent immersed minimal surfaces in $S^{4}$ isometric to it with the same normal curvature function.

浸入圆盘球冠猜想

The spherical cap conjecture for immersed disks

证明了欧氏三维空间中非零常平均曲率且边界映射为圆的浸入圆盘必为嵌入球冠。

On doubly periodic minimal surfaces in $\mathbb H^2 \times \mathbb R$ with finite total curvature in the quotient space

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in $\mathbb H^2$ or a hyperbolic translation along a geodesic in $\mathbb H^2.$ In fact, we prove that if a properly immersed minimal surface in $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2π,$ and…

Gromov-Hausdorff convergence theory of surfaces

In this paper, we use the viewpoint of Gromov-Haustorff convergence to give some new comprehension of well known theorem,it is Huber's classification theorem\cite{Huber}\cite{MS}for complete Riemannian surfaces immersed in $\mathbb{R}^n$ with finite total curvature( $\int_Σ|A|^2<+\infty$) it depend heavily on Müller and Šverák's Hardy-estimate\cite{MS} for the curvature form of surfaces immersed in $\mathbb{R}^n$ with finite total curvature.

具有有界平均曲率的子流形的平均退出时间

Mean exit times from submanifolds with bounded mean curvature

该研究针对微分几何中子流形浸入问题，提出不依赖无穷远弱极大值原理的方法，证明无限平均退出时间的子流形无法等距极小浸入特定乘积空间的几类区域，并推广了随机完备子流形的不可浸入性结果，还给出小平均曲率子流形的矩塔估计。

欧氏空间中浸入高斯曲率收缩子的刚性

Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space

该研究证明，对任意$m\geq2$，满足$K=\langle X,\nu\rangle$的连通闭定向光滑浸入必为单位球面上的嵌入，通过分析极浸入与面积测度完成刚性分类。

具有常接触角的平行平均曲率曲面

Parallel mean curvature surfaces with constant contact angle along free boundaries

本文研究了具有常接触角的平行平均曲率曲面，通过分类和构造例子验证了相关定理的准确性。

Constant curvature hyperspheres and the Euler Characteristic

We show how in many cases the algebraic number of immersed hyperspheres of constant (and prescribed) curvature may be related to the Euler Characteristic of the ambient space.

$\mathbb{R} \times_{f} \mathbb{R}^{2}$ 中具有某些常曲率曲面的高度估计

Height estimates for surfaces with some constant curvature in $\mathbb{r} \times_{f} \mathbb{r}^{2}$

本文利用共形参数推导翘积空间中正外在或平均曲率曲面的必要方程，给出高度估计并分类紧致极小图，扩展了先前研究。

Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov.

A relation between the curvature ellipse and the curvature parabola

At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in $\mathbb R^4$ to $\mathbb R^3$ in a tangent direction corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where m…

Existence and uniqueness of constant mean curvature spheres in Sol_3

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real number…

Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends

We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.

Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bo…

Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$

Codimension two Umbilic points on Surfaces Immersed in R^3

In this paper is studied the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

在三维球面与双曲空间中具有圆形边界的常平均曲率圆盘

Constant mean curvature disks with circular boundary in the $3$-sphere and in hyperbolic space

该文证明三维球面或双曲空间中边界为圆的紧致浸入常平均曲率圆盘必为嵌入的全脐圆盘。

$h$-Principle for Curves with Prescribed Curvature

We prove that every immersed $C^2$-curve $γ$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_γ$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_γ$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary positive curvature in each isotopy class, which generalizes a similar result by McAtee for $C^2$-knots of constant curvature.