Narrow Results

In this article, we obtain the following generalization of isometric C¹-immersion theorem of Nash and Kuiper. Let M be a smooth manifold of dimension m and H a rank k subbundle of the tangent bundle TM with a Riemannian metric g_H. Then the pair (H, g_H) defines a sub-Riemannian structure on M. We call a C¹-map f : (M, H, g_H) → (N, h) into a Riemannian manifold (N, h) a partial isometry if the derivative map df restricted to H is isometric, that is if f*h|_H = g_H. We prove that if f₀ : M → N is a smooth map such that df₀|_H is a bundle monomorphism and , then f₀ can be homotoped to a C¹-map f : M → N which is a partial isometry, provided dim N > k. As a consequence of this result, we obtain that every sub-Riemannian manifold (M, H, g_H) admits a partial isometry in ℝⁿ, provided n ≥ m + k.

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of ${ℂ}^n$, $n\ge 1$, such that the convex hull of $Y$ is all of ${ℂ}^n$. Let ${\mathcal{𝒪}}_{\ast }(X,Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic 1-form $𝜃$ on $X$, not identically zero, and let $\pi :{\mathcal{𝒪}}_{\ast }(X,Y)\to H^1(X,{ℂ}^n)$ send a map $g$ to the cohomology class of $g𝜃$. Our main theorem states that $\pi$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on $X$ can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $u$ with

A theory of simplification of singularities of maps is developed, based on Gromov's convex integration theory. In particular, subject to mild bundle hypotheses, up to a C⁰-small smooth ambient isotopy of embeddings, the singularities induced by the restriction of a submersion to a smoothly embedded closed submanifold are simplified to be only fold i.e. Morse singularities.

Gromov's theorem on directed embeddings of open manifolds is generalized to the case of closed manifolds, where the embeddings are directed with respect to a given codimension ≥ 1 foliation on the manifold. Applications include directed embeddings with prescribed codimension 1 cusp singularities.

Let Ω ⊂ ℝⁿ be a bounded open set. If a sequence f_k : Ω → ℝ^N converges to f in L^∞ in a certain "controlled" manner while bounded in W^1,p (1 < p < + ∞) or BV, we show that f ∈ W^1,p (respectively, f ∈ BV) and ∇f_k → ∇f almost everywhere, where ∇f_k and ∇f are the usual gradients if f_k ∈ W^1,p (respectively, the absolutely continuous part of the gradient measures if f_k ∈ BV). Our main theorem generalizes results for Lipschitz mappings. We show by an example that when p = 1, the limit of a sequence of increasing functions may fail to be in W^1,1 and can even be nowhere C¹.

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any $T_4$ configurations. Here, we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.

We consider entropy solutions to the Cauchy problem for the isentropic compressible Euler equations in the spatially periodic case. In more than one space dimension, the methods developed by De Lellis–Székelyhidi enable us to show here failure of uniqueness on a finite time-interval for entropy solutions starting from any continuously differentiable initial density and suitably constructed bounded initial linear momenta.

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis–Székelyhidi, in particular, we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the $\mathrm{\Lambda }$-convex hull of the constitutive set. An important application of our scheme has been exhibited in recent work by Gallenmüller–Wiedemann.

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d\ge 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T=\text{div}h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u\in W^{1,\tilde{p}}$ and a density $\rho \in L^p$ for which $\text{div}(\rho u)=0$ but $\text{div}(\beta (\rho )u)=T$, provided $1/p+1/\tilde{p}\ge 1+1/(d-1)$.