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Isometric Immersions and Applications

Qing Han
Marta Lewicka

Communicated by Notices Associate Editor Daniela De Silva

1. The Problem of Isometric Immersion and Embedding

The concept of a Riemannian manifold , an abstract -dimensional manifold with a metric structure, was first formulated by Bernhard Riemann in 1868 to generalize classical objects such as curves and surfaces. A manifold is a topological space that locally resembles Euclidean space near each point: in (we write only to emphasize the dimension ) each point has a neighborhood that is homeomorphic to an open subset of . Induced by that homeomorphism, there is a tangent space , namely a -dimensional vector space equipped with a positive-definite inner product . The family is called a Riemannian metric on .

All Euclidean spaces are manifolds and, endowed with the standard Euclidean inner product, are Riemannian manifolds. The simplest nontrivial -dimensional example is the unit sphere consisting of all unit vectors in . At each , the tangent space is the hyperplane in perpendicular to the unit vector . The standard inner product in restricted to induces then a Riemannian metric on , called the round metric. Naturally, there arises the question if any abstract can be identified as a submanifold of some Euclidean space with its induced metric. This is the isometric embedding question, which has assumed a position of fundamental conceptual importance in differential geometry.

The metric requirement can be expressed in terms of partial differential equations (PDEs). Consider a special case: let be the unit ball in , regarded as a -dimensional manifold. The given inner product can be identified with a symmetric, positive definite matrix at each point , so that is a function from to . Isometrically immersing in some Euclidean space means that there exists such that the induced metric agrees with at each point:

Above, we used the matrix notation for the transpose and product. We call the map in 1 an isometric embedding or immersion according to whether it is injective or not. In the general setting, and when globally defined, both sides of 1 become second-order covariant tensors. We now review several important aspects of isometric embeddings and immersions of Riemannian manifolds in the Euclidean space. For details, we refer to 2 and references therein.

2. General Isometric Immersion of Riemannian Manifolds

2.1. Analytic immersions

Let us examine the system 1 closely. First, we notice that it has unknowns and equations, corresponding to the entries in symmetric matrices. The integer is called the Janet dimension. As a rule in solving PDEs, the number of unknowns should be bigger than or equal to the number of equations, otherwise solutions are not expected to exist, in general. Hence we require that . In 1873, Ludwig Schlaefli conjectured that every -dimensional smooth Riemannian manifold admits a smooth local isometric embedding in . It was more than 50 years later that an affirmative answer was given in the analytic case by Maurice Janet (for ) and Élie Cartan (for ).

2.2. Smooth immersions

Second, 1 is classified as a first-order nonlinear system of PDEs, the order of derivatives being and the derivatives of appearing quadratically (nonlinearly). Nonlinear PDEs are solved by Newton’s method, an iteration process generating an actual solution from its approximation. Here, the difficulty arises from the loss of derivatives at each iteration.

To explain this clearly, recall how Newton’s method is used to find a root of a single-variable function. Let be scalar-valued, twice continuously differentiable on . Pick an initial guess for a root of , meaning that is small. To find such that is smaller, we use Taylor’s theorem:

for some . It is natural to choose such that and hence:

We see that is linear in and so is quadratic in . We can iterate this process and obtain a sequence inductively by:

We next need to prove that converges to some and converges to . As a consequence, , resulting in a root of at . An important condition in the convergence proof is:

for some positive constant , independent of . With 2 and an appropriate condition on , we get that indeed converges to zero fast, so that .

We now apply the same idea to the problem 1, which is equivalent to finding the roots of:

We first take some such that is small, write and compute:

where we arranged the expression in the right-hand side according to the powers of (the derivatives of) . As before, we want to choose such that:

This is a first-order linear system of PDEs that can be solved by imposing appropriate boundary values. However, there is a serious issue in the iteration.

In studying PDEs or systems thereof, we need to introduce appropriate spaces of functions and equip them with norms. The process outlined above is performed in the so formed Banach spaces to generate a “root” of the functional . An important feature of 3 is that although a linear combination of derivatives of is governed by , it is , not , that is determined by . More generally, derivatives of of a certain order determine only derivatives of up to the same order. Hence, in the inductive version of 3 and a counterpart of 2:

each step in the iteration contributes a loss of derivatives. If is, say, times continuously differentiable then , as it involves , is times continuously differentiable and thus is also times differentiable and so is . Then, will be continuous only and does not make sense. As a consequence, the iteration process is terminated.

In an outstanding paper published in 1956, John Nash introduced the technique of smoothing operators to compensate for the aforementioned loss of derivatives. He proved that any smooth -dimensional Riemannian manifold admits a (global) smooth isometric embedding in , for in the compact case and in the general case. Specifically for 4, Nash replaced by a function with a better regularity to regain the lost derivative, then solved for , and modified the iteration process accordingly. His technique proved to be extremely useful and it is now known as the hard implicit function theorem, or the Nash-Moser iteration.

2.3. Quest for the smallest dimension

Following Nash, one naturally looks for the smallest dimension of the ambient space. In 1970, Mikhael Gromov and Vladimir Rokhlin, and independently John Greene, proved that any -dimensional smooth Riemannian manifold admits a local smooth isometric embedding in . The proof is based on Nash’s iteration scheme. In his book 1, Gromov studied various related problems; in particular he showed that is enough for the compact case. Then in 1989, Matthias Günther vastly simplified Nash’s original proof: by rewriting the differential equations, he was able to employ the contraction mapping principle instead of the Nash-Moser iteration. Günther also improved the dimension of the target space to . It is still not clear whether this is the best possible result.

For better results are available. According to Gromov and Günther, any compact smooth can be isometrically embedded in smoothly. The local version of this result, with as the target space, is due to Eduard Poznyak in 1973. This also follows from the local existence of solutions to the Darboux equation 6 to be introduced, due to Hartman-Wintner in 1950. Equation 6 and the case of and will be discussed in a separate section below.

The case , is sharply different from the case in which there is only one curvature function, determining the type of the Darboux equation associated to 1; for , the role of various curvatures is not clear. In 1983, Robert Bryant, Phillip Griffiths, and Dean Yang studied the characteristic varieties associated with differential systems for the isometric embedding in of smooth . They proved that these varieties are never empty if , implying, in particular, that the governing systems are never elliptic, no matter what assumptions are put on curvatures. They proved that the characteristic varieties are smooth for and not smooth for . In 2012, the first author of this paper and Marcus Khuri extended that result to any under an additional “smallness” assumption.

For , Bryant, Griffiths, and Yang in 1983 classified types of differential systems for the isometric embedding, by curvature functions. Here, an important quantity is the signature of the curvature tensor viewed as a symmetric linear operator on the space of -forms. In particular, any smooth admits a smooth local isometric embedding in if the signature is different from and . In 1989, Yusuke Nakamura and Yota Maeda showed the same existence result if the curvature tensor is not zero; their key argument was the local existence of solutions to PDEs of principal type. In 2018, Chen, Clelland, Slemrod, Wang, and Yang provided an alternative proof using strongly symmetric positive systems. The result still holds in case the curvature vanishes at one point but its first derivative does not vanish, due to Thomas Poole in 2010.

2.4. Nonsmooth immersions

On the other end of the spectrum, are results showing that isometric immersions have different qualitative behaviour at low and high regularity. Nash in 1954 and Nicolaas Kuiper in 1955 proved the existence of a global isometric embedding of -dimensional Riemannian manifolds in . These results in fact show that every short immersion (or embedding), i.e., for which condition 1 is replaced by:

can be uniformly approximated by -regular actual solutions (immersions or embeddings) to 1. The inequality above is understood pointwise, in the sense of matrices. This abundance of solutions, usually referred to as flexibility results, is typical in applications of Gromov’s h-principle in which a PDE is replaced by a partial differential relation (a differential inclusion) whose solutions are then modified through an iteration technique called convex integration to produce a nearby solution of the underlying PDE.

In 1965, Yuri Borisov used this approach to Hölder - regular solutions and announced that flexibility holds with regularity for any and analytic on . He subsequently gave full details of the proof in 2004 for dimension and . In 2012 Sergio Conti, Camillo De Lellis, and Laszlo Szekelyhidi validated the Borisov’s statement in case of metrics on -dimensional balls, and in case of compact Riemannian manifolds with . The same exponent bounds also hold for any target ambient dimension . On the other hand, in 1978 Anders Källen proved that any metric, with , allows for flexibility up to exponent , provided that is sufficiently large. Recently, the second author of this paper proved flexibility of the related Monge-Ampére system, which is the small slope approximation of the isometric immersion problem 1 around , and in which any subsolution can be uniformly approximated by exact solutions, for any , in agreement with the fully nonlinear case at . We also mention that following Källen’s construction, one can achieve any regularity of solutions to the Monge-Ampére system when .

Dependence of the flexibility threshold exponents on reflects the technical limitation of the method rather than the absolute lack of flexibility beyond those thresholds. In the proofs, the symmetric, positive definite “defect” is decomposed into a linear combination of rank-one defects with nonnegative coefficients. Each of these “primitive defects” is then cancelled by adding to a small but fast oscillating perturbation (a corrugation), which however causes increase of the second derivative of by the factor of the oscillation frequency. The ultimate Hölder regularity of the approximating immersion interpolates between the controlled norms of the adjusted and the blow-up rate of the norm, dictated by the number of these -dimensional adjustments. In case of higher codimension when , several primitive defects may be cancelled at once, reducing the blow-up rate of second derivatives and thus improving the regularity exponent . In the same vein, if the number of primitive defects in the decomposition of can be lowered, for example by an appropriate change of variables, then flexibility holds with higher . This observation is precisely behind the improved regularity statements for -dimensional problems, listed in the next section.

3. Isometric Embeddings of Surfaces

3.1. Local isometric embeddings in

We now give an overview of the question of isometrically embedding a -dimensional Riemannian manifold in . There are basically two methods to study the local case. The first one, already known to Jean Darboux in 1894, restates the problem equivalently as finding a local solution of a nonlinear equation of the Monge-Ampère type. Specifically, let be a metric on a simply connected for some . If there exists for some , satisfying such that:

then admits a isometric immersion in . The equation 6 is called the Darboux equation. Its type is determined by the sign of the Gauss curvature of : elliptic if is positive, hyperbolic if is negative, and degenerate if vanishes. Remarkably, even today, the local solvability of 6 in the general case is not covered by any known theory of PDEs.

A different method to study the local isometric embedding of surfaces in relies on the classical theory of surfaces asserting that such immersion exists provided the solvability of the Gauss-Codazzi system. Namely, let be the Christoffel symbols of the given metric , and its Gauss curvature. Then the coefficients of the second fundamental form satisfy the equations:

We note in passing that the first attempt to establish the local isometric embedding of surfaces in was neither through 6 nor 7: in 1908, Hans Levi solved the case of surfaces with negative curvature by using the equations of virtual asymptotes.

It was several decades later that 6 attracted attention of those interested in the isometric embedding. In the early 1950s, Philip Hartman and Aurel Wintner studied 6 with and proved existence of its local solution. The case when vanishes did not give way to the efforts of mathematicians for a long time. In 1985 and 1986, Chang-Shou Lin made important breakthroughs, establishing existence in a neighborhood of such that and (in 2005 Han gave an alternative proof of this result), or such that in the whole neighbourhood. Later, in 1987, Gen Nakamura covered the case of , and . For the case of nonpositive , Jia-Xing Hong in 1991 also proved the existence of a sufficiently smooth local isometric embedding in a neighborhood of if , where is a negative function and is a function with and . In 2010, the first author of this paper and Khuri proved existence of the smooth local isometric embedding near if changes its sign only along two smooth curves intersecting transversely at . All these results are based on careful studies of the Darboux equation.

In 2003, the first author of this paper, Hong, and Lin studied 7 and proved existence of the local isometric embedding for a large class of metrics with nonpositive Gaussian curvature , for which directional derivative has a simple structure of its zero set. This gives the results of Nakamura and Hong as special cases. On the other hand, Aleksei Pogorelov in 1972 constructed a metric on with a sign-changing such that cannot be realized as a surface in for any . Nikolai Nadirashvili and Yu Yuan in 2008 constructed a metric on of , with no local isometric embedding in .

3.2. Global isometric embeddings in

In 1916, Hermann Weyl posed the following problem: does every smooth metric on with positive Gauss curvature admit a smooth isometric embedding in ? The first attempt to solve it was made by Weyl himself, through the continuity method and the a priori estimates up to the second derivatives. Twenty years later, Hans Lewy solved the problem for analytic metrics . In 1953, Luis Nirenberg gave a complete solution under the very mild hypothesis that is . The result was extended to the case by Erhard Heinz in 1962. In a completely different approach, Aleksandr Alexandroff in 1942 obtained a generalized solution to Weyl’s problem as a limit of polyhedra. The regularity of this generalized solution was proved by Aleksei Pogorelov in the late 1940s. In 1994 and 1995, Pei-Feng Guan and Yanyan Li, and independently Hong and Claude Zuily generalized Nirenberg’s result for metrics with nonnegative Gauss curvature. We also mention that in 1969 Pogorelov showed that if has the Gauss curvature bounded from below by a constant , then it admits a smooth isometric embedding in the 3-dimensional hyperbolic space of curvature .

Closely related to the global isometric embedding problem is the rigidity question. The first rigidity result was proved by Stefan Cohn-Vossen in 1927; this states that any two closed isometric analytic convex surfaces are congruent to each other. His proof was later considerably shortened by O.K. Zhitomirsky. In 1943, Gustav Herglotz gave a very short proof of the rigidity, assuming that the surfaces are three times continuously differentiable. Finally in 1962 this was extended by Richard Sacksteder to surfaces with no more than two times continuously differentiable metrics.

The investigation of the isometric immersion of metrics with negative curvature goes back to David Hilbert. He proved in 1901 that the full hyperbolic plane cannot be isometrically immersed in . The next natural step is to extend such a result to complete surfaces whose Gauss curvature is bounded above by a negative constant. The final solution to this problem was obtained by Nikolai Efimov in 1963: he proved that any complete negatively curved smooth surface does not admit a isometric immersion in if its Gauss curvature is bounded away from zero. In the years following, Efimov extended his result in several ways. Before the 1970s, the study of negatively curved surfaces was largely directed at the nonexistence of isometric immersions in . In the 1980s, Shing-Tung Yau proposed to find a sufficient condition for complete negatively curved surfaces to be isometrically immersed in . In 1993, Hong identified such condition in terms of the Gauss curvature decaying at a certain rate at infinity. His discussion was based on a differential system equivalent to the Gauss-Codazzi system 7.

3.3. Nonsmooth immersions

Extension of the rigidity of Weyl’s problem to Hölder-regular isometric immersions, is originally due to Borisov in the late 1950s, who proved that for , the image of a surface with positive Gauss curvature has bounded extrinsic curvature. Hence, if is a Riemannian manifold with then is the boundary of a bounded convex set that is unique up to rigid motions of , provided that with . In particular, if is constant then . In 2012 Conti, De Lellis, and Szekelyhidi provided a direct analytic proof of this rigidity result.

As we have mentioned before, flexibility for isometric immersions of surfaces in has been proved by Conti, De Lellis, and Szekelyhidi, up to the regularity exponent . This result has been improved by De Lellis, Dominik Inauen, and Szekelyhidi in 2018 where they proved that any short immersion (or embedding) of a -dimensional Riemannian manifold into , can be uniformly approximated by a sequence of isometric immersions (embeddings) for any . Their key argument relies on the fact that every -dimensional metric is locally conformally equivalent to the Euclidean metric . Consequently, a positive definite defect may be, by a change of variables, reduced to the diagonal form, which decomposes into two primitive defects rather than three, resulting in a lower rate of blow-up of the second derivatives in the Nash-Kuiper iteration scheme and subsequently higher regularity of the immersions derived in the limiting process. The same statement, albeit at the linearized level, has been recently used by the second author of this paper, to show density in the space of continuous functions on , of the set of weak solutions with any , to the following equation with a given right-hand side :

For the codimension , this generalizes the prior density result for the Monge-Ampère equation and its weak solutions at , due to Wentao Cao and Szekelyhidi, while the recent improvement by Cao, Jonas Hirsch, and Inauen improve this threshold to . The parallel rigidity statements are likewise available when .

Regarding the flexibility vs rigidity in the regularity interval , Gromov conjectured that the actual threshold occurs sharply at . This is supported by the work of De Lellis and Inauen in 2020 in which they proved that for any , an appropriate convex integration construction yields isometric immersions of a spherical cup, whose Levi-Civita connection differs from the standard one, whereas any such immersion with must necessarily induce the compatible Levi-Civita connection.

4. General Relativity

Quasi-local mass in general relativity is a notion associated with closed spacelike -surfaces in a -dimensional spacetime. Its purpose is to evaluate the amount of matter and gravitational energy within the surface, and can potentially be used to detect the formation of black holes. In this section, we discuss an application of Weyl’s embedding problem to the quasi-local masses; for more information, see 4.

Consider a smooth, orientable, compact Riemannian manifold , with connected boundary of positive Gaussian curvature. According to Weyl’s embedding theorem, may be (uniquely up to rigid motions) isometrically embedded in . This embedding induces the mean curvature , which, in general, differs from the mean curvature of as a submanifold of . In 1992, based on a Hamilton-Jacobi analysis of the Einstein-Hilbert action, David Brown and James York defined the quasi-local mass:

A fundamental result concerning was established by Yuguang Shi and Luen-Fai Tam in 2002. They showed that if is positive and the scalar curvature of is nonnegative, then is nonnegative and it vanishes if and only if isometrically embeds in . From a geometric perspective, this may be interpreted as a comparison theorem for compact manifolds of nonnegative scalar curvature.

Despite this beautiful result, the Brown-York definition has several deficiencies, most notably that it is not “gauge independent” when considered in a spacetime context. This motivated Chiu-Chu Melissa Liu and Shing-Tung Yau in 2003, and then Mu-Tao Wang and Yau in 2009 to each define more general notions of quasi-local mass which satisfy a range of desirable properties. Like , both of these masses also employ the Weyl embedding theorem, and are consequently restricted to surfaces which are topologically -spheres. It should be noted that Wang-Yau utilize the theorem to produce isometric embeddings into Minkowski space, even if the Guassian curvature changes sign. Recently, another quasi-local mass was proposed by Aghil Alaee, Khuri, and Yau, which allows for surfaces of higher genus and also requires their embedding into Minkowski space. A natural question then arises that would have important implications: which closed surfaces admit isometric embeddings into Minkowski space? This problem gives rise to an underdetermined system of equations, and thus one may guess that there are no obstructions. Even for the torus, the problem remains open.

5. Materials Science

When the ambient and intrinsic dimensions agree, , the problem 1 is linked with the orientation preservation by , expressed as:

Under 8, a sufficient and necessary condition for the local solvability of 1 is the vanishing of the Riemannian curvature of , which also guarantees that the solution is smooth and unique up to rigid motions. On the other hand, without the restriction 8, there always exists a Lipschitz continuous constructed by convex integration, that indeed changes orientation in any neighbourhood of any point at which has nonzero curvature. The set of such Lipschitz immersions is dense in the set of short immersions, similarly to other -principle statements that we have listed before.

In the former context, it is natural to pose the quantitative question: what is the infimum of the averaged pointwise deficit of from being an orientation-preserving isometric immersion? This deficit is measured by the non-Euclidean energy on a domain with respect to the Riemannian manifold :

Above, denotes the special orthogonal group of rotations in , and is the distance in the space of matrices . By the polar decomposition theorem, satisfies 1 and 8 if and only if in , which happens precisely when . The follow-up questions now are:

(i)

quantify the infimum of in relation to ,

(ii)

find the structure of minimizers to 9, if they exist,

(iii)

in the limit of becoming -dimensional, study the asymptotic properties of and its minimizers in relation to the curvatures of .

Figure 1.

The first column gathers equivalent conditions for the scaling , in terms of the Riemann curvatures of . The second column provides the asymptotic expansion of the minimizing sequence. The third column gives the form of the -limits in this infinite hierarchy, see 3.

constraint / regularity asymptotic expansion limiting energy
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