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Added a correction in response to Andy's comment.
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Robert Bryant
Robert Bryant
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As Andy says, the answer is 'no': It is known that there is no harmonic map of degree $1$ from the torus to the $2$-sphere. I forget who first observed this. (I expect it's quite early, soon after Eells-Sampson if notAmended after Andy's comment: It's originally due to J. C. Wood in Eells-Sampson's original paperthe early 1970s, but I'm not where I can get to references right nowsee Andy's comment for the exact reference.)

If I have time, I can put in the argument, but the essential outline of the argument is this:

There are two kinds of harmonic maps from the torus to the $2$-sphere. Those that are conformal and those that are not.

If it is conformal, then, up to reversing the orientation on the torus, it is a holomorphic map, and it is well-known that a non-constant holomorphic map from the torus to the $2$-sphere has degree at least 2. (In fact, there is such a holomorphic map of any degree $d\ge 2$.)

If it is not conformal, then a simple calculation shows that the degree of the mapping is zero. (Essentially, one produces an explicit $1$-form on the torus whose differential is the pullback of the area form on the $2$-sphere.)

Thus, there is no harmonic map of degree 1 from the torus to the $2$-sphere.

As Andy says, the answer is 'no': It is known that there is no harmonic map of degree $1$ from the torus to the $2$-sphere. I forget who first observed this. (I expect it's quite early, soon after Eells-Sampson if not in Eells-Sampson's original paper, but I'm not where I can get to references right now.)

If I have time, I can put in the argument, but the essential outline of the argument is this:

There are two kinds of harmonic maps from the torus to the $2$-sphere. Those that are conformal and those that are not.

If it is conformal, then, up to reversing the orientation on the torus, it is a holomorphic map, and it is well-known that a non-constant holomorphic map from the torus to the $2$-sphere has degree at least 2. (In fact, there is such a holomorphic map of any degree $d\ge 2$.)

If it is not conformal, then a simple calculation shows that the degree of the mapping is zero. (Essentially, one produces an explicit $1$-form on the torus whose differential is the pullback of the area form on the $2$-sphere.)

Thus, there is no harmonic map of degree 1 from the torus to the $2$-sphere.

As Andy says, the answer is 'no': It is known that there is no harmonic map of degree $1$ from the torus to the $2$-sphere. I forget who first observed this. (Amended after Andy's comment: It's originally due to J. C. Wood in the early 1970s, see Andy's comment for the exact reference.)

If I have time, I can put in the argument, but the essential outline of the argument is this:

There are two kinds of harmonic maps from the torus to the $2$-sphere. Those that are conformal and those that are not.

If it is conformal, then, up to reversing the orientation on the torus, it is a holomorphic map, and it is well-known that a non-constant holomorphic map from the torus to the $2$-sphere has degree at least 2. (In fact, there is such a holomorphic map of any degree $d\ge 2$.)

If it is not conformal, then a simple calculation shows that the degree of the mapping is zero. (Essentially, one produces an explicit $1$-form on the torus whose differential is the pullback of the area form on the $2$-sphere.)

Thus, there is no harmonic map of degree 1 from the torus to the $2$-sphere.

Source Link
Robert Bryant
Robert Bryant
  • 108.4k
  • 8
  • 342
  • 453

As Andy says, the answer is 'no': It is known that there is no harmonic map of degree $1$ from the torus to the $2$-sphere. I forget who first observed this. (I expect it's quite early, soon after Eells-Sampson if not in Eells-Sampson's original paper, but I'm not where I can get to references right now.)

If I have time, I can put in the argument, but the essential outline of the argument is this:

There are two kinds of harmonic maps from the torus to the $2$-sphere. Those that are conformal and those that are not.

If it is conformal, then, up to reversing the orientation on the torus, it is a holomorphic map, and it is well-known that a non-constant holomorphic map from the torus to the $2$-sphere has degree at least 2. (In fact, there is such a holomorphic map of any degree $d\ge 2$.)

If it is not conformal, then a simple calculation shows that the degree of the mapping is zero. (Essentially, one produces an explicit $1$-form on the torus whose differential is the pullback of the area form on the $2$-sphere.)

Thus, there is no harmonic map of degree 1 from the torus to the $2$-sphere.