Bending surfaces in Riemannian manifoldsWhy is the half-torus rigid?Is a smooth closed surface in Euclidean 3-space rigid?Riemannian surfaces with an explicit distance function?Is a rhombus rigid on a sphere or torus? And generalizationsIsometric embedding a convex cap to render its boundary planarWhy is the half-torus rigid?Geodesic circles on riemannian manifoldsTweetable way to see that Willmore energy is Möbius invariant?Alexandrov angles in Riemannian manifoldsIntrinsic vs Extrinsic geometry of convex surfacesWhat is known about sufficient conditions for the rigidity of a convex surface?Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
Bending surfaces in Riemannian manifolds
Why is the half-torus rigid?Is a smooth closed surface in Euclidean 3-space rigid?Riemannian surfaces with an explicit distance function?Is a rhombus rigid on a sphere or torus? And generalizationsIsometric embedding a convex cap to render its boundary planarWhy is the half-torus rigid?Geodesic circles on riemannian manifoldsTweetable way to see that Willmore energy is Möbius invariant?Alexandrov angles in Riemannian manifoldsIntrinsic vs Extrinsic geometry of convex surfacesWhat is known about sufficient conditions for the rigidity of a convex surface?Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
$begingroup$
Let $S$ be an immersed surface in $mathbbR^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: Sto mathbbR^3$, such that each $s_t$ induces the same metric on $S$ and no $s_t_1$ and $s_t_2$ are related by an isometry of $mathbbR^3$ (i.e. we rule out trivial deformations by one-parameter subgroups of ambient isometries).
1) Do there exist flexible smooth closed surfaces in $mathbbR^3$? In the polyhedral world the answer is yes, with a well-known example of flexible polyhedron by Connelly.
Convex surfaces seem to be rigid, which should be a theorem of Alexandrov or Cauchy. Are there any other global criterions for rigidity/flexibility?
2) Is locally any surface flexible?
3) Can anybody give me an example of a rigid surface with boundary?
mg.metric-geometry riemannian-geometry
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
$endgroup$
|
show 2 more comments
$begingroup$
Let $S$ be an immersed surface in $mathbbR^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: Sto mathbbR^3$, such that each $s_t$ induces the same metric on $S$ and no $s_t_1$ and $s_t_2$ are related by an isometry of $mathbbR^3$ (i.e. we rule out trivial deformations by one-parameter subgroups of ambient isometries).
1) Do there exist flexible smooth closed surfaces in $mathbbR^3$? In the polyhedral world the answer is yes, with a well-known example of flexible polyhedron by Connelly.
Convex surfaces seem to be rigid, which should be a theorem of Alexandrov or Cauchy. Are there any other global criterions for rigidity/flexibility?
2) Is locally any surface flexible?
3) Can anybody give me an example of a rigid surface with boundary?
mg.metric-geometry riemannian-geometry
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
$endgroup$
1
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
1
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
1
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
1
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49
|
show 2 more comments
$begingroup$
Let $S$ be an immersed surface in $mathbbR^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: Sto mathbbR^3$, such that each $s_t$ induces the same metric on $S$ and no $s_t_1$ and $s_t_2$ are related by an isometry of $mathbbR^3$ (i.e. we rule out trivial deformations by one-parameter subgroups of ambient isometries).
1) Do there exist flexible smooth closed surfaces in $mathbbR^3$? In the polyhedral world the answer is yes, with a well-known example of flexible polyhedron by Connelly.
Convex surfaces seem to be rigid, which should be a theorem of Alexandrov or Cauchy. Are there any other global criterions for rigidity/flexibility?
2) Is locally any surface flexible?
3) Can anybody give me an example of a rigid surface with boundary?
mg.metric-geometry riemannian-geometry
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
$endgroup$
Let $S$ be an immersed surface in $mathbbR^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: Sto mathbbR^3$, such that each $s_t$ induces the same metric on $S$ and no $s_t_1$ and $s_t_2$ are related by an isometry of $mathbbR^3$ (i.e. we rule out trivial deformations by one-parameter subgroups of ambient isometries).
1) Do there exist flexible smooth closed surfaces in $mathbbR^3$? In the polyhedral world the answer is yes, with a well-known example of flexible polyhedron by Connelly.
Convex surfaces seem to be rigid, which should be a theorem of Alexandrov or Cauchy. Are there any other global criterions for rigidity/flexibility?
2) Is locally any surface flexible?
3) Can anybody give me an example of a rigid surface with boundary?
mg.metric-geometry riemannian-geometry
mg.metric-geometry riemannian-geometry
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
asked Aug 15 at 7:59
Dmitry KDmitry K
4744 silver badges11 bronze badges
4744 silver badges11 bronze badges
1
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
1
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
1
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
1
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49
|
show 2 more comments
1
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
1
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
1
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
1
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49
1
1
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
1
1
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
1
1
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
1
1
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The standard reference, where the state of the art concerning all of your questions is found:
Q. Han, J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson produced recently proof, but a flaw emerged.
The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.
Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.
Another very important work (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.
Rigidity of the top half of a torus is discussed here, but perhaps without a complete proof having become clear yet.
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f338398%2fbending-surfaces-in-riemannian-manifolds%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The standard reference, where the state of the art concerning all of your questions is found:
Q. Han, J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson produced recently proof, but a flaw emerged.
The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.
Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.
Another very important work (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.
Rigidity of the top half of a torus is discussed here, but perhaps without a complete proof having become clear yet.
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
The standard reference, where the state of the art concerning all of your questions is found:
Q. Han, J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson produced recently proof, but a flaw emerged.
The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.
Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.
Another very important work (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.
Rigidity of the top half of a torus is discussed here, but perhaps without a complete proof having become clear yet.
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
The standard reference, where the state of the art concerning all of your questions is found:
Q. Han, J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson produced recently proof, but a flaw emerged.
The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.
Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.
Another very important work (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.
Rigidity of the top half of a torus is discussed here, but perhaps without a complete proof having become clear yet.
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
The standard reference, where the state of the art concerning all of your questions is found:
Q. Han, J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
There is still no proof that every compact smooth embedded surface in 3-dimensional Euclidean space is rigid. Michael T. Anderson produced recently proof, but a flaw emerged.
The local flexibility of any smooth surface with nonzero or nonnegative Gauss curvature is, I believe, a straightforward consequence of the results in Han and Hong, although they don't state it explicitly.
Han and Hong prove (theorem 8.1.2) that any smooth closed surface, with nonnegative Gauss curvature vanishing on a nowhere dense set, is rigid.
Another very important work (and a very nice one to read) on these questions is the unpublished manuscript of Mohammed Ghomi, which sums up many of the open questions on curves and surfaces in Euclidean space, and particularly the flexibility questions, with a thoughtful collection of references.
Rigidity of the top half of a torus is discussed here, but perhaps without a complete proof having become clear yet.
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
edited Aug 20 at 13:38
BS.
8,0792 gold badges30 silver badges45 bronze badges
8,0792 gold badges30 silver badges45 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
answered Aug 15 at 8:09
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
16k2 gold badges32 silver badges64 bronze badges
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f338398%2fbending-surfaces-in-riemannian-manifolds%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Regarding 3): It is relatively easy to set up boundary value problems for either certain positively curved or negatively curved surfaces that satisfy certain conditions (which would be satisfied on a sufficiently small neighborhood of a point) that imply infinitesimal rigidity. Using the implicit function theorem, these can be extended to local rigidity results (i.e., uniqueness within a set of sufficiently small deformations). There may be global rigidity results for the boundary value problem for positive curvature using results about Monge-Ampère equations.
$endgroup$
– Deane Yang
Aug 15 at 14:25
1
$begingroup$
Regarding 2): If no boundary conditions are imposed, then the theory of elliptic and hyperbolic PDEs with the implicit function theorem can be used to show that a surface is flexible in a sufficiently small neighborhood of a point with nonzero Gauss curvature
$endgroup$
– Deane Yang
Aug 15 at 14:28
1
$begingroup$
Regarding 1): I believe there is essentially nothing known outside the convex case. I have wondered whether it might be possible to smooth Connelly's polyhedral example by replacing the vertices by a smooth surface that flexes in the appropriate way.
$endgroup$
– Deane Yang
Aug 15 at 14:31
1
$begingroup$
1 was previously asked here: mathoverflow.net/questions/1975/…
$endgroup$
– j.c.
Aug 15 at 17:47
$begingroup$
Not an answer, but for non compact surfaces, see the notion of associated family for a minimal surface en.m.wikipedia.org/wiki/Associate_family giving some of the most famous examples. This idea generalizes to many symmetric spaces other than Euclidean space.
$endgroup$
– Andy Sanders
Aug 15 at 18:49