Roughly speaking, the Almgren-Schoen-Simon paper shows that for the area-minimizing hypersurfaces in well-behaved normed spaces, the singular set have vanishing Hausdorff codimension 2 measure. (Note that for quadratic norm, the singular set is of codimension at least 7). Thus, it is imaginable that for general normed areas on Euclidean space, the Simons cone/failure of Bernstein problems should exist in lower dimensions. It is indeed the case by Frank Morgan, who shows that the cone over the Clifford torus is area-minimizing with respect to an $SO(2)\times SO(2)$-invariant normed area in $R^4$. This is sharp in view of the regularity result by Almgren-Schoen-Simon. Then the work of Mooney and Yang gives counterexamples to Bernstein-type theorems, similar in spirit to the connection mentioned by Ian Agol for the quadratic norm.

Remark 1: Here is an anecdote I heard from William Allard. Before Simons' paper on stable cones, Wendell Fleming once said that heuristically with dimension increasing, the portion of area inside a unit ball is decreasing. Thus, it's conceivable that with the dimension increasing, strange things should happen.

Remark 2: Notably, by Gary Lawlor's classical results on vanishing calibrations (https://bookstore.ams.org/memo-91-446/) a minimal cone of arbitrary codimension is area-minimizing as long as the second fundamental form of its link is relatively small compared to its dimension. Thus, when dimensions increase, area-minimizing cones should become more and more abundant in some sense, indeed verifying Fleming's heuristic. Lawlor's criterion accurately detects all area-minimizing hypercones known up to date, and can even prove the instability in cases of lower dimensional Simons/Lawson cones. Almost all non-special-holonomic area-minimizing cones known up to today, i.e., excluding holomorphic, special Lagrangians, associatives, etc., either can be proven to be minimizing using Lawlor's criterion or are precisely discovered this way.

Roughly speaking, the Almgren-Schoen-Simon paper shows that for the area-minimizing hypersurfaces in well-behaved normed spaces, the singular set has vanishing Hausdorff codimension 2 measure. (Note that for quadratic norm, the singular set is of codimension at least 7). Thus, it is imaginable that for general normed areas on Euclidean space, the Simons cone/failure of Bernstein problems should exist in lower dimensions. It is indeed the case by Frank Morgan, who shows that the cone over the Clifford torus is area-minimizing with respect to an $SO(2)\times SO(2)$-invariant normed area in $R^4$. This is sharp in view of the regularity result by Almgren-Schoen-Simon. Then the work of Mooney and Yang gives counterexamples to Bernstein-type theorems, similar in spirit to the connection mentioned by Ian Agol for the quadratic norm.

Remark 1: Here is an anecdote I heard from William Allard. Before Simons' paper on stable cones, Wendell Fleming once said that heuristically with dimension increasing, the portion of area inside a unit ball is decreasing. Thus, it's conceivable that with the dimension increasing, strange things can happen.

Remark 2: Notably, by Gary Lawlor's classical results on vanishing calibrations (https://bookstore.ams.org/memo-91-446/) a minimal cone of arbitrary codimension is area-minimizing as long as the second fundamental form and the cotangent of the focal radius of its link are relatively small compared to its dimension. Thus, when dimensions increase, area-minimizing cones should become more and more abundant in some sense, indeed verifying Fleming's heuristic. Lawlor's criterion accurately detects all area-minimizing hypercones known up to date, and can even prove the instability in cases of lower dimensional Simons/Lawson cones. Almost all non-special-holonomic area-minimizing cones known up to today, i.e., excluding holomorphic, special Lagrangians, associatives, etc., either can be proven to be minimizing using Lawlor's criterion or are precisely discovered this way.

At least for the cases of the Bernstein problem for minimal graphs, Simons' theorem on stable minimal cones, etc., there is some evidence that dimension 8 is more likely a coincidence rather than having deep intrinsic reasons.

Roughly speaking, the Almgren-Schoen-Simon paper shows that for the area-minimizing hypersurfaces in well-behaved normed spaces, the singular set have vanishing Hausdorff codimension 2 measure. (Note that for quadratic norm, the singular set is of codimension at least 7). Thus, it is imaginable that for general normed areas on Euclidean space, the Simons cone/failure of Bernstein problems should exist in lower dimensions. It is indeed the case by Frank Morgan, who shows that the cone over the Clifford torus is area-minimizing with respect to a $SO(2)\times SO(2)$-invariant normed area in $R^4$. This is sharp in view of the regularity result by Almgren-Schoen-Simon. Then the work of Mooney and Yang give counterexamples to Bernstein-type theorems, similar in spirit to the connection mentioned by Ian Agol for the quadratic norm.

I think for such normed areas which are $C^\infty$ close to the quadratic area, such phenomena cannot happen, thus retaining the magical 8. However, it is conceivable that one should be able to find fairly symmetric norms so that the corresponding magical number is any of 8, 7, 6, 5, 4. Thus, the number 8, in this case, might not be as magical as one thinks, if one sees it in a more general setting, say Finsler geometry.

At least for the cases of the Bernstein problem for minimal graphs, Simons' theorem on stable minimal cones, etc., there is some evidence that dimension 8 is more likely a coincidence rather than having deep intrinsic reasons. In more general settings, that 8 is as magical as 7,6,5, or 4.

Roughly speaking, the Almgren-Schoen-Simon paper shows that for the area-minimizing hypersurfaces in well-behaved normed spaces, the singular set have vanishing Hausdorff codimension 2 measure. (Note that for quadratic norm, the singular set is of codimension at least 7). Thus, it is imaginable that for general normed areas on Euclidean space, the Simons cone/failure of Bernstein problems should exist in lower dimensions. It is indeed the case by Frank Morgan, who shows that the cone over the Clifford torus is area-minimizing with respect to an $SO(2)\times SO(2)$-invariant normed area in $R^4$. This is sharp in view of the regularity result by Almgren-Schoen-Simon. Then the work of Mooney and Yang gives counterexamples to Bernstein-type theorems, similar in spirit to the connection mentioned by Ian Agol for the quadratic norm.

I think for such normed areas which are $C^\infty$ close to the quadratic area, such phenomena cannot happen, thus retaining the magical 8. However, it is conceivable that one should be able to find fairly symmetric norms so that the corresponding magical number is any of 8, 7, 6, 5, 4. Thus, the number 8, in this case, might not be as magical as one thinks, if one sees it in a more general setting, say Finsler geometry.

Remark 1: Here is an anecdote I heard from William Allard. Before Simons' paper on stable cones, Wendell Fleming once said that heuristically with dimension increasing, the portion of area inside a unit ball is decreasing. Thus, it's conceivable that with the dimension increasing, strange things should happen.

Remark 2: Notably, by Gary Lawlor's classical results on vanishing calibrations (https://bookstore.ams.org/memo-91-446/) a minimal cone of arbitrary codimension is area-minimizing as long as the second fundamental form of its link is relatively small compared to its dimension. Thus, when dimensions increase, area-minimizing cones should become more and more abundant in some sense, indeed verifying Fleming's heuristic. Lawlor's criterion accurately detects all area-minimizing hypercones known up to date, and can even prove the instability in cases of lower dimensional Simons/Lawson cones. Almost all non-special-holonomic area-minimizing cones known up to today, i.e., excluding holomorphic, special Lagrangians, associatives, etc., either can be proven to be minimizing using Lawlor's criterion or are precisely discovered this way.